Miscellaneous Problems

Find the integral $\int \frac{x^{2}+x-1}{x^{3}-x^{2}-x+1}\, dx$.

Find the integral $\int \frac{x\, dx}{(x-1)^{3}(x-2)^{2}}$.

Find the integral $\int \frac{x^{5}\, dx}{(x-1)^{2}(x^{2}-1)}$.

Find the integral $\int \frac{x^{2}\, dx}{(x-a)(x-b)(x-c)}$.

Find the integral $\int \frac{dx}{x^{5}+2x^{3}+3x^{2}}$.

Find the integral $\int \frac{x^{2}-1}{x^{4}-x^{2}+1}\, dx$.

Find the integral $\int \frac{x^{2}(1-x^{2})\, dx}{(1+x^{2})^{3}}$.

Find the integral $\int \frac{x^{2}\, dx}{x^{4}+x^{2}-2}$.

Find the integral $\int \frac{x\, dx}{(x^{2}+2x+2)^{2}}$.

Find the integral $\int \frac{x^{9}\, dx}{(x^{2}-1)^{3}}$.

Find the integral $\int \frac{x^{3}+3x}{(x^{2}-1)(x^{2}+1)^{2}}\, dx$.

Find the integral $\int x\sqrt{\frac{a-x}{a+x}}\, dx$.

Find the integral $\int \frac{x\, dx}{1+\sqrt{x-1}}$.

Find the integral $\int \frac{dx}{\sqrt{2x+1}-\sqrt{x}}$.

Find the integral $\int \frac{x+\sqrt{x+1}}{1+\sqrt{x+1}}\, dx$.

Find the integral $\int \frac{x^{3}\, dx}{\sqrt{2ax-x^{2}}}$.

Find the integral $\int \frac{x\, dx}{(a+bx)^{\frac{3}{2}}}$.

Find the integral $\int \frac{dx}{x^{2}(a+bx^{2})^{\frac{3}{2}}}$.

Find the integral $\int \frac{dx}{x^{3}\sqrt{x-1}}$.

Find the integral $\int \frac{dx}{x^{\frac{3}{4}}\left(1+x^{\frac{1}{3}}\right)}$.

Find the integral $\int \frac{dx}{x^{2}\sqrt{ax+bx^{4}}}$.

Find the integral $\int \frac{x^{3}\, dx}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}$.

Find the integral $\int \frac{dx}{(1-\sqrt{1-x^{2}})^{2}}$.

Find the integral $\int \frac{x^{3}\, dx}{(1+\sqrt{x})\sqrt{x-x^{2}}}$.

Find the integral $\int \frac{dx}{(2x-1)\sqrt{1-x^{2}}}$.

Find the integral $\int \frac{x^{3}\, dx}{(1+x^{2})^{2}\sqrt{1-x^{2}}}$.

Find the integral $\int \frac{dx}{(x+1)\sqrt{x^{2}+x+1}}$.

Find the integral $\int \frac{\sqrt{a+bx^{2}}}{x^{6}}\, dx$.

Find the integral $\int \frac{dx}{x^{2}\sqrt{x^{2}-x}}$.

Find the integral $\int \frac{x^{3}\, dx}{(x-1)^{2}\sqrt{x^{2}+1}}$.

Find the integral $\int \frac{dx}{x^{4}\sqrt{x^{2}+1}}$.

Find the integral $\int \frac{(2-x^{3})\, dx}{(1+x^{3})\sqrt{1+x^{3}}}$.

Find the integral $\int \frac{x+1}{x-2}\frac{dx}{\sqrt{1+x^{3}}}$.

Find the integral $\int \frac{dx}{x\sqrt{1-x^{2}-x^{4}}}$.

Find the integral $\int x\sqrt{\frac{1-x^{2}}{1+x^{2}}}\, dx$.

Find the integral $\int \frac{(x^{2}-a^{2})\, dx}{x\sqrt{a^{4}+x^{4}}}$.

Find the integral $\int \frac{dx}{(1+x^{4})\sqrt{\sqrt{1+x^{4}}-x^{2}}}$.

Find the integral $\int \frac{dx}{\sqrt{(x-2)(2-3x+x^{3})}}$.

Find the integral $\int \frac{3x^{2}+5x^{6}}{\sqrt{1+x^{4}}}\, dx$.

Find the integral $\int \frac{dx}{\sqrt{x^{2}-a^{2k-2}x^{2k}}}$.

Find the integral $\int \frac{(x^{4}-a^{4})\, dx}{(x^{4}+a^{4})\sqrt{x^{4}+a^{4}}}$.

Find the integral $\int \frac{1+x^{2}}{1-2x^{2}\cos \alpha +x^{4}}\, dx$.

Find the integral $\int \frac{x^{2}\, dx}{(x\sin x+\cos x)^{2}}$.

Find the integral $\int \frac{x^{2}\, dx}{(\sin x-x\cos x)^{2}}$.

Find the integral $\int \frac{dx}{(a+(ax+b)\tg x)^{2}}$.

Find the integral $\int \frac{a+x}{x^{2}+ax\ln x}\, dx$.

Find the integral $\int \frac{1-\ln x}{(x+a\ln x)^{2}}\, dx$.

Find the integral $\int \frac{1+x\ln x}{x}e^{x}\, dx$.

Find the integral $\int \left(\frac{1-x}{1+x^{2}}\right)^{2}e^{x}\, dx$.

Find the integral $\int \frac{1+x^{2}}{(1+x)^{2}}e^{x}\, dx$.

Find the integral $\int e^{\arctg x}(1+2x)\, dx$.

Find the integral $\int e^{\sqrt{x}}x\, dx$.

Find the integral $\int \frac{xe^{x}}{(e^{x}-1)^{3}}\, dx$.

Find the integral $\int \frac{dx}{\cos x-2\tg x}$.

Find the integral $\int \frac{dx}{\sin^{4}x\cos x}$.

Find the integral $\int \frac{dx}{1-\cos^{4}x}$.

Find the integral $\int \frac{dx}{\sin x\cos^{2}x}$.

Find the integral $\int \frac{\ln \tg x\, dx}{\cos^{4}x}$.

Find the integral $\int \frac{\cos^{2}x\, dx}{\sin x(1+a^{2}\cos^{2}x)}$.

Find the integral $\int \frac{\cos^{2}x\, dx}{3+\sin x}$.

Find the integral $\int \frac{dx}{\sin x\sqrt{\cos 2x}}$.

Find the integral $\int \tg x\sqrt{\cos 2x}\, dx$.

Find the integral $\int \frac{dx}{\sqrt{2+3\sin x+\sin^{2}x}}$.

Find the integral $\int \frac{\sqrt{\sin x}}{1+\sin^{2}x}\, dx$.

Find the integral $\int (b^{2}-a^{2}\tg^{2}x)^{\frac{3}{2}}\, dx$.

Find the integral $\int \frac{dx}{\cos x\sqrt{\cos 2x}}$.

Find the integral $\int \arctg \sqrt{x}\, dx$.

Find the integral $\int \frac{\arctg x}{(1-x^{2})\sqrt{1-x^{2}}}\, dx$.

Find the integral $\int e^{a\cdot \arctg x}\frac{x\, dx}{(1+x^{2})\sqrt{1+x^{2}}}$.

Find the integral $\int e^{\sin x}\sin 2x\, dx$.

Find the integral $\int e^{x}\sin^{2}x\, dx$.

Find the integral $\int \frac{x+\sin x}{1+\cos x}\, dx$.

Find the integral $\int \frac{(x+1)\ln (x+1)}{\sqrt{x^{2}+2x+3}}\, dx$.

Find the integral $\int \ln (\sqrt{x-a}+\sqrt{x-b})\, dx$.

Find the integral $\int \frac{dx}{2^{x}+1}$.

Find the integral $\int \frac{x+1}{2^{x}}\, dx$.

Find the integral $\int \frac{\sin x\, dx}{\sqrt{\cos^{3}2x}}$.

Find the integral $\int \frac{\cos x\, dx}{\sqrt{\cos^{3}2x}}$.

Find the integral $\int \frac{dx}{\sin x\sqrt[4]{\cos 2x}}$.

Find the integral $\int \sin^{n-1}x\cos (n+1)x\, dx$.

Find the integral $\int \sin^{n-1}x\sin (n+1)x\, dx$.

Find the integral $\int \frac{dx}{a+b\cos x+c\sin x}$.

Find the integral $\int \frac{\sqrt{\ctg x}-\sqrt{\tg x}}{1+3\sin 2x}\, dx$.

Find the integral $\int x^{\sqrt{2}}\sin \ln x\, dx$.

Find the integral $\int \frac{\ln (\cos \varphi +\sqrt{\cos 2\varphi })}{1-\cos^{2}\varphi }\, d\varphi$.

Find the integral $\int \frac{1+\sin x}{1+\cos x}e^{x}\, dx$.

Find the integral $\int \frac{\sin x-\cos x}{\sin^{2}x}e^{x}\, dx$.

Find the integral $\int \frac{\sin x}{\sin (x-\alpha )}\, dx$.

Find the integral $\int \frac{a\sin x+b\cos x}{\sqrt{1+\sin x\cos x}}\, dx$.

Find the integral $\int \frac{dx}{\cos x\cos (x-\alpha )}$.

Find the integral $\int \frac{dx}{\sqrt{a+b+(a+2b)\cos x+b\cos 2x}}$.

Find the integral $\int \frac{dx}{\cos^{3}x+\sin^{3}x}$.

Find the integral $\int \frac{dx}{\cos^{5}x-\sin^{5}x}$.

Find the integral $\int x^{2}\sin^{3}x\, dx$.