Integrals sqrt(a^2+u^2), a 0

Question 1

$\int\sqrt{a^2+u^2}du$

Accepted Answer

$= \frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln(u+\sqrt{a^2+u^2})+C$

Question 2

$\int{u^2\sqrt{a^2+u^2}du}$

Accepted Answer

$= \frac{u}{8}(a^2+2u^2)\sqrt{a^2+u^2}-\frac{a^4}{8}\ln(u+\sqrt{a^2+u^2})+C$

Question 3

$\int\frac{\sqrt{a^2+u^2}}{u}du$

Accepted Answer

$= \sqrt{a^2+u^2}-a\ln\left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C$

Question 4

$\int\frac{\sqrt{a^2+u^2}}{u^2}du$

Accepted Answer

$= -\frac{\sqrt{a^2+u^2}}{u}+\ln(u+\sqrt{a^2+u^2})+C$

Question 5

$\int\frac{du}{\sqrt{a^2+u^2}}$

Accepted Answer

$= \ln(u+\sqrt{a^2+u^2})+C$

Question 6

$\int\frac{u^2du}{\sqrt{a^2+u^2}}$

Accepted Answer

$\frac{u}{2}\sqrt{a^2+u^2}-\frac{a^2}{2}\ln(u+\sqrt{a^2+u^2})+C$

Question 7

$\int\frac{du}{u\sqrt{a^2+u^2}}$

Accepted Answer

$= -\frac{1}{a}\ln\left|\frac{\sqrt{a^2+u^2}+a}{u}\right|+C$

Question 8

$\int\frac{du}{u^2\sqrt{a^2+u^2}}$

Accepted Answer

$= -\frac{\sqrt{a^2+u^2}}{a^2u}+C$

Question 9

$\int\frac{du}{(a^2+u^2)^{3/2}}$

Accepted Answer

$= \frac{u}{a^2\sqrt{a^2+u^2}}+C$

Integrals sqrt(a^2+u^2), a > 0