Learn Integrals with Examples and Practice Problems

I have in trouble for evaluating following integral
$\int_{0}^{\infty} (\sqrt{1 + x^{4}} - x^{2}) d x = \frac{Γ^{2} (\frac{1}{4})}{6 \sqrt{π}}$

piarepm 2022-01-05

Integrate:
$\int \frac{x^{2}}{(x \cos x - \sin x) (x \sin x + \cos x)} d x$

James Dale 2022-01-05

Prove:
$\int_{0}^{\infty} x^{2 n} e^{- x^{2}} d x = \frac{(2 n)!}{2^{2 n} n!} \frac{\sqrt{π}}{2}$

Mary Reyes 2022-01-05

How to compute
$\int_{- \infty}^{\infty} \exp (- \frac{{(x^{2} - 13 x - 1)}^{2}}{611 x^{2}}) d x$

Roger Smith 2022-01-05

Can this integral be solved with contour integral or by some application of residue theorem?
$\int_{0}^{\infty} \frac{\log (1 + x)}{1 + x^{2}} d x = \frac{π}{4} \log 2 + Catalan constant$

Adela Brown 2022-01-05

Evaluate $\int \frac{1}{{(x^{2} + 1)}^{2}} d x$

James Dale 2022-01-05

Evaluate the integral:
$\int_{0}^{1} (\sqrt[3]{1 - x^{7}} - \sqrt[7]{1 - x^{3}}) d x$

Miguel Reynolds 2022-01-05

What is the proof of the following:
$\int_{0}^{1} {(\frac{\ln t}{1 - t})}^{2} d t = \frac{π^{2}}{3}$ ?

Adela Brown 2022-01-05

Evaluate
$\int_{0}^{2 π} \frac{1}{{(1 + a \cos θ)}^{2}} d θ, 0 \leq a < 1$

Algotssleeddynf 2022-01-05

I cannot prove that if $f (x)$ is convex on $[a, b]$ then
$f (\frac{a + b}{2}) \leq \frac{1}{b - a} \int_{a}^{b} f (x) d x \leq \frac{f (a) + f (b)}{2}$

Russell Gillen 2022-01-04

How do I show that:
$\int_{0}^{\infty} \frac{\sin (a x) \sin (b x)}{x^{2}} d x = π min \frac{a, b}{2}$

Nicontio1 2022-01-04

Evaluate the integral:
$\int \sin (x^{3}) d x$

Francisca Rodden 2022-01-04

I want to evaluate the following integral via complex analysis
$\int_{x = 0}^{x = \infty} e^{- a x} \cos (b x) d x, a > 0$
Which function/ contour should I consider ?

Algotssleeddynf 2022-01-04

I have to compute this integral
$\int_{0}^{\infty} \frac{d t}{1 + t^{4}}$

osnomu3 2022-01-04

Which is the easiest way to evaluate $\int_{0}^{\frac{π}{2}} (\sqrt{\tan x} + \sqrt{\cot x})$ ?

David Young 2022-01-04

Some time ago I came across to the following integral:
$I = \int_{0}^{1} \frac{1 - x}{1 + x} \frac{d x}{\ln x}$
What are the hints on how to compute this integral?

Cynthia Bell 2022-01-04

Solving limits:
$f (x) = lim_{n \to \infty} (n \int_{0}^{\frac{π}{4}} {(\tan x)}^{n} d x)$

untchick04tm 2022-01-04

Any tips on how to show that
$\int_{0}^{π} f (\sin x) d x = 2 \int_{0}^{\frac{π}{2}} f (\sin x) d x$

b2sonicxh 2022-01-04

How do I integrate the following:
$\int \frac{1 + x^{2}}{(1 - x^{2}) \sqrt{1 + x^{4}}} d x$

Joanna Benson 2022-01-04

I would like to show that
$\int_{0}^{1} \frac{x - 1}{\ln (x)} d x = \ln 2$

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Calculus 1

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