Integral Questions with Solutions: Practice and Master Integration

The Cauchy's Beta Integral is given by
$\int_{- \infty}^{\infty} \frac{d t}{(1 + i t)^{x} (1 - i t)^{y}} = \frac{π 2^{2 - x - y} Γ (x + y - 1)}{Γ (x) Γ (y)}$

Fillipus P Kasita2022-11-06

If f : [0, ∞) → R is continuous, the Laplace transform of f is the function F defined by F(s) = Z ∞ 0 f(x)e −sx dx and the domain of F is the set consisting of all numbers for which the integral converges. Find the Laplace transforms of the following functions. a) f(x) = 10 b) f(x) = sin 2x c) f(x) = e x d) f(x) =

Kayden Mills 2022-11-06

Show that $\int_{0}^{π / 2} \frac{\log^{2} \sin x \log^{2} \cos x}{\cos x \sin x} d x = \frac{1}{4} (2 ζ (5) - ζ (2) ζ (3))$

Emmanuel Umar2022-11-03

Obtain the integral of the function $y = \int x c o s x d x$

podvelkaj8 2022-10-31

Integral check $\int \frac{6 x + 4}{x^{4} + 3 x^{2} + 5} dx$

ajanlr 2022-10-30

Evaluating $\int \frac{1}{x^{7} - x} d x$

Valery Cook 2022-10-29

About $\int_{0}^{π / 2} \arctan (1 - \sin^{2} (x) \cos^{2} (x)) d x = π (\frac{π}{4} - \arctan \sqrt{\frac{\sqrt{2} - 1}{2}})$

Aryanna Blake 2022-10-29

Need help solving complicated integral $\int e^{- x} \cos 4 x \cos 2 x d x$

Jairo Decker 2022-10-24

Can you please explain this equality?
$\int \frac{r}{σ^{2}} \exp (\frac{- r^{2}}{2 σ^{2}}) d r = - \exp (\frac{- r^{2}}{2 σ^{2}})$

Christopher Saunders 2022-10-23

Compute the indefinite integral
$\int \frac{7 x^{13} + 5 x^{15}}{(x^{7} + x^{2} + 1)^{3}} d x$

snaketao0g 2022-10-23

Use the residues theorem to calulate
$\int_{| z - i | = 3} \frac{e^{z^{2}} - 1}{z^{3} - i z^{2}} d z$

Kamila Frye 2022-10-22

Is it true that $\int t \frac{d F}{d \ln t} d \ln t = \int \frac{d F}{d t} d t$

Marilyn Cameron 2022-10-20

What is
$\int_{1}^{3} \frac{e^{\frac{1}{x}}}{x^{2}} d x ?$
How do you approach solving this?

Deborah Proctor 2022-10-18

If:
$λ = \int_{0}^{1} \frac{d x}{1 + x^{3}};$
Then evaluate:
$p = lim_{n \to \infty} {(\frac{\prod_{r = 1}^{n} (n^{3} + r^{3})}{n^{3 n}})}^{1 / n};$

mafalexpicsak 2022-10-16

Show that :
$\int_{0}^{\infty} \frac{e^{- x}}{x^{2}} {(\frac{1}{1 - e^{- x}} - \frac{1}{x} - \frac{1}{2})}^{2} d x = \frac{7}{36} - \ln A + \frac{ζ (3)}{2 π^{2}}$

klastiesym 2022-10-16

Calculate $\int 5^{x + 1} e^{2 x - 1} d x$

propappeale00 2022-10-16

Solve the integral
$\int \frac{1 + x + \sqrt{1 + x^{2}}}{\sqrt{x + 1} + \sqrt{x}} d x$

Chaim Ferguson 2022-10-15

How do you show that
$\int_{0}^{\infty} \frac{\log | \tan x |}{1 + x^{2}} d x = \frac{π}{2} \log (\tanh 1) ?$

Nathalie Case 2022-10-15

Evaluate the intergral using the fundamental theorem of calculus.
$\int_{- 10}^{10} (8 x^{9} + 5 x^{5}) d x$

Deja Bradshaw 2022-10-14

Suppose f(x) is such that
$\int_{- \infty}^{\infty} e^{t x} f (x) d x = \arcsin (t - \sqrt{\frac{1}{2}})$

Although solving integrals starts in high school, these answers that are provided are related to post-secondary problems that are approached by Data Science students, economists, and engineers in their daily tasks. It’s a reason why integral questions that you will see below will span across a wide range of problems. You can also compare your integral problem and see equations that relate to your question. Alternatively, it will help you with solving integrals step by step. There are many methods to do that as the provided answers show. Starting with solving integrals with substitution to variables, the help is there.

Integral Calculus

Integral Problem Solving Made Easy: Learn with Examples