Learn – Calculus – Improper Integrals

1 Type 1
- 1.1 Exercises
2 Type 2
- 2.1 Exercises
  - 2.1.1 Problem set
  - 2.1.2 Problem set
3 Convergence tests
- 3.1 Exercises

Type 1

Exercises

Problem set

In each of the following problems, evaluate the integral. In case the integral diverges, say so.

$\int\limits_1^{\infty} \frac{1}{x^2}\, dx$

$\int\limits_1^{\infty} \frac{1}{x^3}\, dx$

$\int\limits_1^{\infty} \frac{1}{x}\, dx$

$\int\limits_1^{\infty} \frac{1}{\sqrt{x}}\, dx$

$\int\limits_1^{\infty} \frac{1}{\sqrt[3]{x}}\, dx$

$\int\limits_1^{\infty} \frac{1}{x^{1.01}}\, dx$

Problem set

In each of the following problems, evaluate the integral. In case the integral diverges, say so.

$\int\limits_1^{\infty} \frac{1}{e^x}\, dx$

$\int\limits_1^{\infty} \sin x\, dx$

$\int\limits_0^{\infty} \frac{1}{1+x^2}\, dx$

$\int\limits_1^{\infty} \frac{\ln x}{x^2}\, dx$

$\int\limits_0^{\infty} \frac{1}{x+5} - \frac{1}{x+9}\, dx$

$\int\limits_0^{\infty} \frac{1}{x^2+3x+2}\, dx$

$\int\limits_0^{\infty} \frac{3}{x^2+4x+5}\, dx$

$\int\limits_2^{\infty} te^{-t}\, dt$

Problem set

Evaluate the following.

$\int\limits_{-\infty}^{-1} \frac{1}{x^2}\, dx$

$\int\limits_{-\infty}^{-1} \frac{1}{x}\, dx$

$\int\limits_{-\infty}^0 \frac{1}{x+5} - \frac{1}{x+9}\, dx$

$\int\limits_{-\infty}^{\infty} e^{-|x|}\, dx$

Type 2

Exercises

Problem set

Evaluate the following.

$\int\limits_0^1 \frac{1}{\sqrt{x}}\, dx$

$\int\limits_0^1 \frac{1}{x}\, dx$

$\int\limits_0^1 \frac{1}{x^2}\, dx$

$\int\limits_0^{\frac{\pi}{2}} sec^2 x\, dx$

$\int\limits_0^1 \ln x\, dx$

Problem set

Evaluate the following.

$\int\limits_{-1}^1 \frac{1}{\sqrt[5]{x}}\, dx$

$\int\limits_2^4 \frac{1}{\sqrt[3]{x-3}}\, dx$

Convergence tests

Exercises

For each of the following problems, check if the integral converges or diverges by comparing with an appropriately chosen function.

$\int\limits_{1}^{\infty} e^{-x^2}\, dx$