Improper Integrals of Unbounded Functions

Compute the integral or establish its divergence. $\int_{0}^{1} \frac{d x}{\sqrt{x}}$.

Compute the integral or establish its divergence. $\int_{-2}^{0} \frac{d x}{(x+1) \sqrt[3]{x+1}}$.

Compute the integral or establish its divergence. $\int_{0}^{4} \frac{d x}{\sqrt{x}+x}$.

Compute the integral or establish its divergence. $\int_{0}^{2} \frac{d x}{x \sqrt{x}-2 x+\sqrt{x}}$.

Compute the integral or establish its divergence. $\int_{0}^{2} \frac{d x}{\sqrt{\left|x^{2}-1\right|}}$.

Compute the integral or establish its divergence. $\int_{0}^{3} \frac{x^{2} d x}{\sqrt{9-x^{2}}}$.

Compute the integral or establish its divergence. $\int_{0}^{0,5} \frac{d x}{x \ln^{2} x}$.

Compute the integral or establish its divergence. $\int_{0}^{1} \frac{d x}{x \ln^{2} x}$.

Compute the integral or establish its divergence. $\int_{0}^{\pi } \operatorname {tg} x d x$.

Compute the integral or establish its divergence. $\int_{0}^{\pi / 4} \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}} d x$.

Compute the integral or establish its divergence. $\int_{0}^{\pi } \frac{\left|\cos x\right|}{\sqrt{\sin x}} d x$.

Compute the integral or establish its divergence. $\int_{0}^{e} \frac{d x}{e^{x}-1}$.

Compute the integral or establish its divergence. $\int_{-1}^{0} e^{1 / x} \frac{d x}{x^{3}}$.

Compute the integral or establish its divergence. $\int_{-1}^{1} e^{1 / x} \frac{d x}{x^{3}}$.

Compute the integral or establish its divergence. $\int_{0}^{1} \frac{\arcsin x}{\sqrt{1-x^{2}}} d x$.

Compute the integral or establish its divergence. $\int_{0}^{1} \frac{d x}{\sqrt{\left(1-x^{2}\right) \arcsin x}}$.

Compute the integral. $\int_{0}^{1} \frac{2-\sqrt[3]{x}-x^{3}}{\sqrt[5]{x^{3}}} d x$.

Compute the integral. $\int_{-0,5}^{-0,25} \frac{d x}{x \sqrt{2 x+1}}$.

Compute the integral. $\int_{\sqrt{2}}^{2} \frac{d x}{(x-1) \sqrt{x^{2}-2}}$.

Compute the integral. $\int_{1}^{2} \frac{d x}{x \sqrt{3 x^{2}-2 x-1}}$.

Compute the integral. $\int_{-1}^{1} \frac{d x}{(4-x) \sqrt{1-x^{2}}}$.

Compute the integral. $\int_{-1}^{1} \frac{d x}{\left(16-x^{2}\right) \sqrt{1-x^{2}}}$.

Compute the integral. $\int_{-1}^{1} \frac{x^{4} d x}{\left(1+x^{2}\right) \sqrt{1-x^{2}}}$.

Compute the integral. $\int_{-a}^{a} \frac{d x}{\sqrt{a^{2}+b^{2}-2 b x}}, a > 0, b \geqslant 0$.

Compute the integral. $\int_{0}^{1} \frac{x^{n} d x}{\sqrt{1-x^{2}}}, n \in \mathbb {N}$.

Compute the integral. $\int_{a_{2}}^{b} x \sqrt{\frac{x-a}{b-x}} d x, b > a$.

Compute the integral. $\int_{0}^{\pi / 4} \sqrt{\operatorname {ctg} x} d x$.

Compute the integral. $\int_{0}^{\pi / 2} \sqrt{\operatorname {tg} x} d x$.

Compute the integral. $\int_{0}^{2}\left(x \sin \frac{\pi }{x^{2}}-\frac{\pi }{x} \cos \frac{\pi }{x^{2}}\right) d x$.

Compute the integral. $\int_{-1}^{1} \frac{d x}{\sqrt{\left(1-x^{2}\right) \arccos x}}$.

Compute the integral. $\int_{0}^{1} \frac{x^{3} \arcsin x}{\sqrt{1-x^{2}}} d x$.

Compute the integral. $\int_{-1}^{1} x^{3} \ln \frac{1+x}{1-x} \frac{d x}{\sqrt{1-x^{2}}}$.

Compute the integral. $\int_{0}^{1} x^{\alpha } \ln^{n} x d x, \quad \alpha > -1, \quad n \in \mathbb {N}$.

Compute the integral. $\int_{0}^{\pi / 2} \ln \cos x d x$.

Compute the integral. $\int_{0}^{\pi } x \ln \sin x d x$.

Compute the integral. $\int_{0}^{\pi / 2}(\ln \cos x) \cos (2 n x) d x, n \in \mathbb {N}$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{1}{\sqrt{2-5 x}}, x \in [0 ; 0,4)$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{x+1}{\sqrt[5]{x^{3}}}, x \in [-1 ; 1], x \neq 0$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{x}{\sqrt{(x-a)(b-x)}}, x \in (a ; b)$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{x^{2}}{\sqrt{(x-3)(5-x)}}, x \in (3 ; 5)$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{1}{x \sqrt{\ln x}}, x \in (1 ; e]$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = x \ln \frac{1+x}{1-x}, x \in [0 ; 1)$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{\arcsin \sqrt{x}}{\sqrt{1-x}}, x \in [0 ; 1)$.

Compute the area of the curvilinear trapezoid formed by the graph of the function. $y = \frac{\left|\ln x\right|}{\sqrt{1-x^{2}}}, x \in (0 ; 1)$.

Find the area of the figure bounded by the given curve and its asymptote. $x y^{2} = 8-4 x$.

Find the area of the figure bounded by the given curve and its asymptote. $(x+1) y^{2} = x^{2}, x < 0$.

Find the area of the figure bounded by the given curve and its asymptote. $(4-x) y^{2} = x^{3}$.

Find the area of the figure bounded by the given curve and its asymptote. $\left(1-x^{2}\right) y^{2} = x^{2}, x > 0$.

Find the area of the figure bounded by the given curve and its asymptote. $x = \cos 2 t, y = \cos 2 t \operatorname {tg} t, t \in [\pi / 4 ; 3 \pi / 4]$.

Find the area of the figure bounded by the curves given in polar coordinates:

Find the volume of the solid bounded by the surface obtained by rotating the curve $y = e^{-x^{2}}$ and the line $y = 0$ about the ordinate axis.

Find the volume of the solid bounded by the surface obtained by rotating the curve $(4-x) y^{2}-x^{3} = 0$ about its asymptote.

Prove the inequality. $\frac{\pi }{10} < \int_{0}^{2} \frac{d x}{(4+\sqrt{\sin x}) \sqrt{4-x^{2}}} < \frac{\pi }{8}$.

Prove the inequality. $\frac{\ln (2+\sqrt{3})}{11} < \int_{1}^{2} \frac{d x}{(10+\sin x) \sqrt{x^{2}-1}} < \frac{\ln (2+\sqrt{3})}{10}$.

Prove the inequality. $\int_{1,9}^{2} \frac{e^{-x} d x}{\sqrt[4]{2+x-x^{2}}} < 0,03$.

Prove the inequality. $\int_{0}^{1} \frac{\sin (\pi / 4-x)}{\sqrt{1-x^{2}}} d x > 0$.

Investigate the integral for convergence. $\int_{0}^{8} \frac{d x}{x^{2}+\sqrt[3]{x}}$

Investigate the integral for convergence. $\int_{0}^{2} \sqrt{\frac{16+x^{4}}{16-x^{4}}} d x$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{dx}{\sqrt[5]{1 - x^{10}}}$.

Investigate the integral for convergence. $\int_1^2 \frac{(x-2) d x}{x^3-3 x^2+4}$.

Investigate the integral for convergence. $\int_{0}^{\pi } \frac{\sin x}{x^{2}} d x$.

Investigate the integral for convergence. $\int_0^{2 \pi } \frac{d x}{\sqrt[3]{\sin x}}$.

Investigate the integral for convergence. $\int_1^\pi \sin \left(\frac{1}{\cos x}\right) \frac{d x}{\sqrt{x}}$.

Investigate the integral for convergence. $\int_{-\pi / 4}^{\pi / 4} \sqrt{\frac{\cos x-\sin x}{\cos x+\sin x}} d x$.

Investigate the integral for convergence. $\int_{1}^{3} \frac{d x}{\sqrt{\operatorname {tg}\left(x^{3}-7 x^{2}+15 x-9\right)}}$.

Investigate the integral for convergence. $\int_{-1}^{1} \frac{d x}{\ln (1+x)}$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{d x}{\sqrt[3]{x\left(e^{x}-e^{-x}\right)}}$.

Investigate the integral for convergence. $\int_{0}^{2} \frac{\sqrt{x} d x}{e^{\sin x}-1}$.

Investigate the integral for convergence. $\int_0^\pi \frac{\operatorname {sh} x d x}{e^{x^2}-\cos x}$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{d x}{\sqrt{x}+\operatorname {arctg} x}$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{\ln x}{\sqrt{x(1-x)^{3}}} dx$.

Investigate the integral for convergence. $\int_{0}^{\pi } \frac{\ln x}{\sqrt{\sin x}} dx$.

Investigate the integral for convergence. $\int_{0}^{\pi } \frac{\ln \sin x}{\sqrt[3]{x}} d x$.

Investigate the integral for convergence. $\int_{0}^{\pi } \frac{\ln \sin x}{x \sqrt{\sin x}} d x$.

Investigate the integral for convergence. $\int_{0}^{1} \ln \left|1-4 \sin^{2} x\right| d x$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{\arcsin \left(x^{2}+x^{3}\right)}{x \ln^{2}(1+x)} d x$.

Investigate the integral for convergence. $\int_{0}^{1} \frac{d x}{\arccos x}$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{\pi } \frac{1-\cos x}{x^{\alpha }} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{6 x^{2 x^{2}}+24 \cos x-13 x^{4}-30}{\sin^{\alpha } x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} e^{\alpha / x}(\cos x)^{1 / x^{3}} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{e^{\alpha x}-\sqrt{1+x}}{\operatorname {ch} x-\cos x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\sqrt{e^{2}+x^{2}}-e^{\cos x}}{x^{\alpha }} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{\pi / 2} \frac{\cos^{2} 2 x-e^{-4 x^{2}}}{x^{\alpha } \operatorname {tg} x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{\pi / 2} \frac{e^{\alpha \cos x}-\sqrt{1+2 \cos x}}{\sqrt{\cos^{5} x}} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\operatorname {ch}(\alpha x)-\ln \left(1+x^{2}\right)-1}{\sqrt[3]{8-x^{3}}-2} \cdot d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\ln \left(e^{x^{\alpha }}+x\right)-x}{\operatorname {tg} x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{0,5} \frac{\ln \operatorname {tg} x d x}{(4 x \cos x-\pi \sin x)^{\alpha }}$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\ln^{\alpha } \operatorname {ch}(1 / x)}{\ln^{3}(1+x)} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\ln \sqrt{1+2 x}-x e^{-x}}{1-\cos^{\alpha } x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{-1}^{1}\left(\frac{1+x}{1-x}\right)^{\alpha } \ln (2+x) d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{1}^{2} \frac{\operatorname {arctg}(x-1)}{(x-\sqrt{x})^{\alpha }} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\sin \left(\arcsin x+x^{3}\right)-x}{\sin^{\alpha } x} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{(1-x)^{5 / 3} d x}{\operatorname {arctg} \alpha \left(x-x^{2}\right)}$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{\operatorname {arctg}\left(x^{2}+x^{2 \alpha }\right)}{x \ln^{\alpha }(1+x)} d x$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} \frac{d x}{\ln \left|x-\alpha \right|}$.

Find all values of the parameter $\alpha$ for which the integral converges. $\int_{0}^{1} e^{x^{2} /(x-a)} d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{1} x^{\alpha }(1-x)^{\beta } d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{\pi / 2} \sin^{\alpha } x \cos^{\beta } x d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{1} x^{\alpha } \ln^{\beta } \frac{1}{x} d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{1} x^{\alpha }(1-x)^{\beta } \ln x d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{0,5} \frac{\ln^{\alpha }(1 / x)}{\operatorname {tg}^{\beta } x} d x$.

Determine for which values of the parameters $\alpha$ and $\beta$ the integral converges. $\int_{0}^{\pi } \frac{\sin^{\alpha -1} x d x}{(1+\beta \cos x)^{\alpha }}, \beta \geqslant 0$.

Investigate the integral for absolute and conditional convergence. $\int_{0}^{1} \frac{\sin (1 / x) d x}{x^{2}+\sqrt{x^{3}}+x^{2} \cos (1 / x)}$.

Investigate the integral for absolute and conditional convergence. $\int_{0}^{1} \frac{1}{x \sqrt{x}} \cos \frac{\sqrt{x}-1}{\sqrt{x}} d x$.

Investigate the integral for absolute and conditional convergence. $\int_{0}^{1}\left(1-e^{\sqrt[3]{x^{2}} \cos (1 / x)}\right) \frac{d x}{x^{2}}$.

Investigate the integral for absolute and conditional convergence. $\int_{0}^{0,5} \frac{\cos^{3} \ln x}{x \ln x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1}(1-x)^{\alpha } \sin \frac{\pi }{1-x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{x^{\alpha }}{x^{2}+1} \sin \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \cos \left(\frac{1}{\sqrt{x}}-1\right) \frac{d x}{x^{\alpha }}$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{0,5}\left(\frac{x}{1-x}\right)^{\alpha } \cos \frac{1}{x^{2}} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\cos (1 / x) d x}{x^{2}(1 / x+\sin (1 / x))^{\alpha }}$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{\pi / 4} \sin \left(\frac{1}{\sin x}\right) \frac{d x}{\sin^{\alpha } x}$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{\pi / 4} \operatorname {tg}^{\alpha } x \cos \operatorname {ctg} x d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{-1}^{1} \sin \frac{1+x}{1-x} \frac{d x}{\left(1-x^{2}\right)^{\alpha }}$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{x^{\alpha }}{e^{x}-1} \sin \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} x^{\alpha } \operatorname {arctg} x \cos \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\sin x^{\alpha }}{x^{2}} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\sin (1 / x)}{(\sqrt{x}-x)^{\alpha }} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{(1-x)^{\alpha }}{x} \sin \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameters $\alpha$ and $\beta$

Find all values of the parameter $\alpha$ for which the integral converges

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1 / 2} \frac{x^{-4} \sin (1 / x)}{\left(1+x^{-2}\right)^{\alpha }} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{e^{\alpha / x} \cos \left(1 / x^{2}\right)}{x^{2}\left(\frac{1}{x^{2}} e^{1 / x}-\ln x\right)^{\alpha }} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\sin x^{\alpha } \cos x^{-\alpha }}{x^{3 / 2}} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\ln^{\alpha }\left(1+x^{2}\right)}{x^{4}} \cos \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{1} \frac{\left(\operatorname {arctg} x^{2}\right)^{\alpha }}{x^{3}} \cdot \sin \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{\pi / 2} \operatorname {tg}^{\alpha } x \sin \frac{1}{x} d x$.

Investigate the integral for absolute and conditional convergence for all values of the parameter $\alpha$. $\int_{0}^{\pi / 2} \operatorname {ctg}^{\alpha } x \cos \frac{1}{x} d x$.

Prove that if the function $f(x)$ is monotonic and bounded on the interval $(0 ; 1]$ and there exists the improper integral

Determine whether a convergent improper integral $\int_{a}^{b} f(x) d x$ can be regarded as the limit of the corresponding integral sum

Let the function $f(x)$ be monotonic on the interval $(a ; b]$ and unbounded in the right neighborhood of the point $x = a$. Prove that if $\int_{a}^{b} f(x) d x$ exists, then

Prove that if the function $f(x)$ is continuous on some interval $[0 ; \varepsilon ), \varepsilon > 0$, and $f(0) \neq 0$, then for $\alpha < 1$ the asymptotic equality is valid

Find the integral in the principal-value sense. v.p. $\int_{0}^{10} \frac{d x}{7-x}$.

Find the integral in the principal-value sense. v.p. $\int_{-1}^{7} \frac{d x}{(x-1)^{3}}$.

Find the integral in the principal-value sense. v.p. $\int_{a}^{b} \frac{d x}{(x-c)^{n}}, c \in (a ; b), n \in N$.

Find the integral in the principal-value sense. v.p. $\int_{0.5}^{4} \frac{d x}{x \ln x}$.

Find the integral in the principal-value sense. v.p. $\int_{0}^{\pi } x \operatorname {tg} x d x$.

Find the integral in the principal-value sense. v.p. $\int_{0}^{\pi / 2} \frac{d x}{3-5 \sin x}$.

Find the integral in the principal-value sense. v.p. $\int_{0}^{\pi / 2} \frac{d x}{\alpha -\sin x}, \alpha \in (0 ; 1)$.

For which values of $\alpha$ does v.p. $\int_{0}^{2} \frac{x^{\alpha }}{1-x} d x$ exist?

Prove that if the function $f(x)$ is continuous on the segment $[a ; b]$ and vanishes only at the single point $x = c \in (a ; b)$, the derivative $f^{\prime }(x)$ exists in some neighborhood of the point $x = c$, with $f^{\prime }(c) \neq 0$, and $f^{\prime \prime }(c)$ exists, then $\int_{a}^{b} \frac{d x}{f(x)}$ diverges, while v.p. $\int_{a}^{b} \frac{d x}{f(x)}$ exists.

Prove that for $x > 1$ the value v.p. $\int_{0}^{x} \frac{d t}{\ln t}$ exists.