Improper integrals

Compute

$${\int }_2^{+\infty }\frac{dx}{x^4}.$$

Solution available to those who have unlocked the subject.

Compute

$${\int }_0^{+\infty }{e}^{-3x}dx.$$

Solution available to those who have unlocked the subject.

Compute

$${\int }_0^{+\infty }x{e}^{-x^2}dx.$$

Solution available to those who have unlocked the subject.

Compute

$${\int }_0^1\frac{dx}{x^{\frac{2}{3}}}.$$

Solution available to those who have unlocked the subject.

Compute

$${\int }_0^1\frac{dx}{\sqrt{1-x}}.$$

Solution available to those who have unlocked the subject.

Compute

$${\int }_0^{\frac{1}{2}}\frac{dx}{x{\ln }^{2}x}.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_1^{+\infty }\frac{x^2+1}{x^4+x+1}dx.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_1^{+\infty }\frac{dx}{x+\sqrt{x}}.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_0^{+\infty }\frac{dx}{x^{\frac{1}{3}}(1+x)}.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_0^1\frac{e^{2x}-1}{x^2}dx.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_0^1\frac{1-\cos x}{x^{\frac{7}{2}}}dx.$$

Solution available to those who have unlocked the subject.

Determine for which $\alpha \in \mathbb{R}$ the following integral converges:

$${\int }_1^{+\infty }\frac{dx}{x^{\alpha }(1+x^2)}.$$

Solution available to those who have unlocked the subject.

Determine for which $\alpha \in \mathbb{R}$ the following integral converges:

$${\int }_0^1\frac{e^x-1}{x^{\alpha }}dx.$$

Solution available to those who have unlocked the subject.

Determine the behavior of

$${\int }_1^{+\infty }\frac{\cos x}{x^3}dx.$$

Solution available to those who have unlocked the subject.