Learn Integrals with Examples and Practice Problems

Integral $\int_{0}^{y} d x \frac{\sin (a x + \frac{b}{x})}{x}$

Alvin Pugh 2022-01-24

Integers n satsifying $\frac{1}{\sin \frac{3 π}{n}} = \frac{1}{\sin \frac{5 π}{n}}$

Aubrey Hendricks 2022-01-23

calculate this integral:
$\int_{0}^{\frac{π}{2}} \arccos (\sin x) d x$
the answer is $- d \frac{{π}^{2}}{8}$
It doesnt

Vanessa Hensley 2022-01-23

Evaluating $\int \frac{d x}{\sqrt{4 - 9 x^{2}}}$ with different trig substitutions (sin vs cos) gives different results
For the first trig sub, I set $9 x^{2} = 4 \cos^{2} θ$ . This simplifies to: $x = \frac{23}{\cos θ}$ , and $d x = - \frac{23}{\sin θ}$ . Substituting in, I get:
$\int \frac{- 2 \sin θ}{6 \sin θ} = - \frac{θ}{3} = - \frac{1}{3} \cos^{- 1} (\frac{3 x}{2}) + C$
For the second trig sub, I set $9 x^{2} = 4 \sin^{2} θ$ . This simplifies to: $x = \frac{23}{\sin θ}$ , and $d x = \frac{23}{\cos θ}$ . Substituting in, I get:
$\int \frac{2 \cos θ}{6 \cos θ} = \frac{θ}{3} = {\frac{13}{\sin}}^{- 1} (\frac{3 x}{2}) + C$ (2)
Why do these two trig substitutions yield different results graphically? Shouldn't they result in the same graph?

amevaa0y 2022-01-23

integral of $\sec x \tan x$

Michael Maggard 2022-01-07

Integrate: $\int \frac{3 \sin (x) + 2 \cos (x)}{2 \sin (x) + 3 \cos (x)} d x$

oliviayychengwh 2022-01-07

Compute the following integral:
$\int_{0}^{\infty} \frac{e^{x} \sin (x)}{x} d x$

abreviatsjw 2022-01-07

How to evaluate the integral
$\int e^{x^{3}} d x$

William Cleghorn 2022-01-06

Prove that:
$\int_{0}^{1} \frac{x \ln (1 + x)}{1 + x^{2}} d x = \frac{π^{2}}{96} + \frac{\ln^{2} 2}{8}$

elvishwitchxyp 2022-01-06

How to evaluate the following improper integral :
$\int_{0}^{+ \infty} \frac{x \sin x}{x^{2} + 1} d x$

Michael Maggard 2022-01-06

Evaluate the following integral:
$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} d x$

David Lewis 2022-01-06

Evaluate $\int_{0}^{1} {(\frac{1}{\ln x} + \frac{1}{1 - x})}^{2} d x$

kramtus51 2022-01-06

Consider the following integral:
$I = \int_{0}^{\infty} \frac{x - 1}{\sqrt{2^{x} - 1} \ln (2^{x} - 1)} d x$

Wanda Kane 2022-01-06

Evaluate integral:
$\int_{0}^{\infty} \frac{\ln x}{1 + x^{2}} d x$

compagnia04 2022-01-05

Solve the integral:
$\int_{0}^{2} \frac{d x}{\sqrt{x} (x - 1)}$

Sapewa 2022-01-05

Evaluate the integral $\int_{- \infty}^{\infty} \frac{\cos (x)}{x^{2} + 1} d x$

Helen Lewis 2022-01-05

How to calculate following integration?
$\int (\sqrt{\tan x} + \sqrt{\cot x}) d x$

Donald Johnson 2022-01-05

Evaluation of the integral:
$\int_{0}^{1} \frac{\ln (1 - x)}{1 + x} d x$

Salvatore Boone 2022-01-05

Compute
$\int \frac{x^{2} d x}{{(x \sin x + \cos x)}^{2}}$

Ikunupe6v 2022-01-05

I am in the middle of a problem and having trouble integrating the following integral:
$\int_{- 1}^{1} \frac{1}{{(1 + x^{2})}^{2}} d x$

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Calculus 1

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