Learn Integrals with Examples and Practice Problems

Calculate:
$\int_{0}^{\infty} \frac{\ln^{2} (x)}{{(1 - x^{2})}^{2}} d x$

Irvin Dukes 2022-01-02

How would I go about evaluating this integral?
$\int_{0}^{\infty} \frac{\ln (x^{2} + 1)}{x^{2} + 1} d x$

Algotssleeddynf 2022-01-02

How to compute the following integral?
$\int \log (\sin x) d x$

Deragz 2022-01-01

I need help calculating two integrals
1) $\int_{1}^{2} \sqrt{4 + \frac{1}{x}} d x$
2) $\int_{0}^{\frac{2}{π}} x^{n} \sin (x) d x$

Joseph Krupa 2022-01-01

How to calculate the value of the integrals
$\int_{0}^{1} {(\frac{\arctan x}{x})}^{2} d x$ ,
$\int_{0}^{1} {(\frac{\arctan x}{x})}^{3} d x$
and $\int_{0}^{1} \frac{\arctan^{2} x \ln x}{x} d x$ ?

Ashley Bell 2022-01-01

I want to solve these Integrals
1. $\int_{0}^{\frac{π}{2}} \frac{1}{1 + \tan^{\sqrt{2}} x} d x$
2. $\int_{0}^{\frac{π}{2}} \frac{1}{{(\sqrt{2} \cos^{2} x + \sin^{2} x)}^{2}}$

Wanda Kane 2022-01-01

How would you compute the following integrals?
$I_{n} = \int_{0}^{π} \frac{1 - \cos n x}{1 - \cos x} d x$
$J_{n, m} = \int_{0}^{π} \frac{x^{m} (1 - \cos n x)}{1 - \cos x} d x$

Ernest Ryland 2022-01-01

Sovle integral:
$\int_{0}^{\infty} \log (1 + x^{2}) \frac{cosh \frac{π x}{2}}{{sinh}^{2} \frac{π x}{2}} d x = 2 - \frac{4}{π}$

maduregimc 2021-12-31

Prove that:
$\int_{0}^{\frac{π}{2}} \ln^{2} (\cos x) d x = \frac{π}{2} \ln^{2} 2 + \frac{π^{3}}{24}$

Arthur Pratt 2021-12-31

Is there a general formula for finding the primitive of
$e^{a x} \sin (b x)$

Brock Brown 2021-12-31

It is possible to prove that
$\int_{0}^{\infty} \frac{e^{i x} - e^{- x}}{x} d x = - i \frac{π}{2}$

eiraszero11cu 2021-12-31

I am looking for a short proof that
$\int_{0}^{\infty} {(\frac{\sin x}{x})}^{2} d x = \frac{π}{2}$
What do you think?

piarepm 2021-12-31

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$\int_{0}^{\infty} \frac{\cos x}{1 + x^{2}} d x$

compagnia04 2021-12-31

How to prove
$\int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2}$

Krzychau1 2021-12-31

Show that $\int_{1}^{2} \frac{e^{x} (x - 1)}{x (x + e^{x})} d x = \ln (\frac{2 + e^{2}}{2 + 2 e})$ and $\int_{0}^{\frac{π}{2}} \frac{x \sin (2 x)}{1 + {\cos (2 x)}^{2}} d x = \frac{π^{2}}{16}$

kuhse4461a 2021-12-30

How can you evaluate
$\int_{0}^{\frac{π}{2}} \log \cos (x) d x$ ?

piarepm 2021-12-30

Evaluate the integral $\int_{0}^{\frac{π}{2}} \frac{\sin^{3} x}{\sin^{3} x + \cos^{3} x} d x$

Lennie Davis 2021-12-30

How to evaluate:
$\int_{- \infty}^{\infty} \frac{\ln (x^{2} + 1)}{x^{2} + 1} d x$

zakinutuzi 2021-12-30

Solve integrals using residue theorem?
$\int_{0}^{π} \frac{d θ}{2 + \cos θ}$
$\int_{0}^{\infty} \frac{x}{{(1 + x)}^{6}} d x$

Roger Smith 2021-12-30

I have to calculate this integrals
$I_{1} = \int_{0}^{2} \frac{\arctan (x)}{x^{2} + 2 x + 2} d x$
$I_{2} = lim_{n \to \infty} \int_{0}^{n} \frac{\arctan x}{x^{2} + x + 1} d x$
I have a hint on the first one, the substitution $x = \frac{2 - t}{1 + 2 t}$ , but I can't understand how it was conceived.

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