Math Problems Vault

Problem 5223

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {dx}{1+x}$ , taking the number of subintervals $n=4$ .

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Problem 5224

Compute by the trapezoidal method the integral $\displaystyle \int_{1}^{3} \dfrac {dx}{x+1}$ , taking the number of subintervals $n=4$ .

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Problem 5225

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {dx}{1+x^{2}}$ , taking the number of subintervals $n=4$ .

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Problem 5226

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{\pi /2} \dfrac {\sin x}{x}\, dx$ , taking the number of subintervals $n=4$ .

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Problem 5227

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {\ln (1+x)}{1+x^{2}}\, dx$ , taking the number of subintervals $n=6$ .

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Problem 5228

Compute by the trapezoidal method the integral $\displaystyle \int_{0}^{1} \dfrac {\ln (1+x)}{x}\, dx$ , taking the number of subintervals $n=6$ .

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Problem 5229

Compute the same integrals as in problems 5223--5228, for $n=4$ , by Simpson's formula.

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Problem 5230

Compute the same integrals as in problems 5223--5228 by Cotes's formula.

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Problem 5231

Compute the same integrals as in problems 5223--5228 by Chebyshev's method.

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Problem 5232

Compute the same integrals as in problems 5223--5228 by Gauss's method.

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Problem 5233

After the substitution $x = \sin \varphi$ , compute by Simpson's method with $n=4$ the integrals:

\int _{0}^{1} \frac{x^{2}\, dx}{\sqrt{1-x^{2}}}; \qquad \int _{0}^{1} \frac{dx}{\sqrt{1-x^{4}}}.

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Problem 5234

Compute by Simpson's method with $n=4$ the integrals $\displaystyle \int_{0}^{1} \frac{x^{2}\, dx}{\sqrt{1-x^{2}}}$ and $\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1-x^{4}}}$ with the substitution $1-x=u^{2}$ .

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Problem 5235

After the substitution $x = \dfrac {1}{t}$ , compute by Simpson's method with $n=4$ the integral $\displaystyle \int_{1}^{\infty } \frac{x^{2}\, dx}{\sqrt{1+x^{4}}}$ .

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Problem 5236

Using Simpson's formula and the parametric equation of the ellipse, obtain the approximate formula for the length of the arc of a quarter of an ellipse with semi-axes $a$ and $b$ :

l = \frac{\pi }{12}\left(a+b+\sqrt{8(a^{2}+b^{2})}\right).

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Problem 5237

Investigate the degree of precision of the preceding formula (problem 5236) for $b=0$ , for $b=a$ , and for $b = \dfrac {a}{2}$ (in the last case, compare with the result of Simpson's formula for $n=4$ ).

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Problem 5238

Prove the following Gauss-type quadrature formula:

\int _{-1}^{1} \frac{f(x)\, dx}{\sqrt{1-x^{2}}} = \frac{\pi }{n} \sum _{\nu =1}^{n} f\left(\cos \frac{(2\nu -1)\pi }{2n}\right).

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Problem 5239

Prove the following Gauss-type quadrature formula:

\int _{-1}^{1} \sqrt{1-x^{2}}\, f(x)\, dx = \frac{\pi }{n} \sum _{\nu =1}^{n} \sin ^{2}\frac{\nu \pi }{n}\, f\left(\cos \frac{\nu \pi }{n}\right).

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Problem 5240

Prove the following Gauss-type quadrature formula (setting $m=2n+1$ ):

\int _{-1}^{1} \sqrt{\frac{1-x}{1+x}}\, f(x)\, dx = \frac{4\pi }{m}\sum _{\nu =1}^{n}\sin ^{2}\frac{\nu \pi }{m}\, f\left(\cos \frac{2\pi \nu }{m}\right).

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Problem 5241

Prove the following Gauss-type quadrature formula:

\int _{0}^{1} \frac{f(x)\, dx}{\sqrt{x(1-x)}} = \frac{\pi }{n}\sum _{\nu =1}^{n} f\left(\cos ^{2}\frac{(2\nu -1)\pi }{4n}\right).

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Problem 5242

Prove the following Gauss-type quadrature formula:

\int _{0}^{1} \sqrt{x(1-x)}\, f(x)\, dx = \frac{\pi }{4n}\sum _{\nu =1}^{n-1}\sin ^{2}\frac{\nu \pi }{n}\, f\left(\cos ^{2}\frac{\nu \pi }{n}\right).

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Problem 5243

Prove the following Gauss-type quadrature formula (setting $m=2n+1$ ):

\int _{0}^{1} \sqrt{\frac{1-x}{x}}\, f(x)\, dx = \frac{2\pi }{m}\sum _{\nu =1}^{n}\sin ^{2}\frac{\nu \pi }{m}\, f\left(\cos ^{2}\frac{\nu \pi }{m}\right).

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Remark.

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Formulas 5239 and 5241 are completely exact if $f(x)$ is a polynomial of degree less than $2n-2$ . The remaining formulas of this section (5238, 5240, 5242, 5243) are exact if $f(x)$ is a polynomial of degree less than $2n$ .

Problem 5244

Using formula 5242 (see also the remark to problem 5243) with $n=4$ , compute the integral $\displaystyle \int_{0}^{1} \sqrt{x(1-x^{2})}\, dx$ .

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Problem 5245

Using formula 5240 (see also the remark to problem 5243) with $n=4$ , compute the integral $\displaystyle \int_{-1}^{1} \frac{dx}{\sqrt{1-x^{4}}}\, dx$ .

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Problem 5246

Prove the exactness of the formula

\int _{0}^{\infty } e^{-x} f(x)\, dx = \frac{2-\sqrt{2}}{4}f\left(2+\sqrt{2}\right) + \frac{2+\sqrt{2}}{4}f\left(2-\sqrt{2}\right),

where $f(x)$ is a polynomial of degree less than four.

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Problem 5247

Prove the exactness of the formula

\int _{-\infty }^{\infty } e^{-x^{2}} f(x)\, dx = \frac{\sqrt{\pi }}{6}\left(f\left(-\sqrt{\frac{3}{2}}\right) + 4f(0) + f\left(\sqrt{\frac{3}{2}}\right)\right),

where $f(x)$ is a polynomial of degree less than six.

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Problem 5248

Apply Gregory's formula to compute the integral $\displaystyle \int_{0}^{1} \frac{dx}{1+x^{2}}$ , taking $n=1$ , $h=0{,}1$ , carrying out the computation with seven decimal digits.

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Remark.

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Gregory's formula (1670): $\dfrac {1}{h}\displaystyle \int_{a}^{a+nh} y\, dx = \dfrac {1}{2}y_{0} + y_{1} + \ldots + \dfrac {1}{2}y_{n} + \sum_{\nu } A_{\nu }\left(\Delta^{\nu }y_{n-\nu } + (-1)^{\nu }\Delta^{\nu }y_{0}\right)$ , where $A_{1} = -\dfrac {1}{12}$ , $A_{2} = -\dfrac {1}{24}$ , $A_{3} = -\dfrac {19}{720}$ , $A_{4} = -\dfrac {3}{160}$ , $A_{5} = -\dfrac {863}{60\, 480}, \ldots$

Problem 5249

The same for Gauss's formula.

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Remark.

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Gauss's formula: $\dfrac {1}{h}\displaystyle \int_{a}^{a+nh} y\, dx = \dfrac {1}{2}y_{0} + y_{1} + \ldots + \dfrac {1}{2}y_{n} + \sum_{\nu } B_{\nu }\left(\dfrac {\Delta^{2\nu +1}y_{n-\nu } + \Delta^{2\nu +1}y_{n-\nu -1}}{2} - \dfrac {\Delta^{2\nu +1}y_{-\nu } + \Delta^{2\nu +1}y_{-\nu -1}}{2}\right)$ , where $B_{0} = -\dfrac {1}{12}$ , $B_{1} = \dfrac {11}{720}$ , $B_{2} = -\dfrac {191}{60\, 480}$ .

Problem 5250

Carry out an analogous computation (using the Gregory and Gauss formulas, problems 5248--5249) for the integral $\displaystyle \int_{100}^{110} \frac{dx}{x}$ .

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