Calculus/Integration/Solutions

Solutions to Set Two

1. $\frac{1}{13}{x}^{13}+C$
2. $2x^4+C$
3. $\frac{4}{3}{x}^3+\frac{11}{4}{x}^4+C$
4. $\frac{31}{33}{x}^{33}+x^4-\frac{9}{5}{x}^5+C$
5. $-5x^{-1}+C$

Solutions to Set Three

1. $\int (\cos x+\sin x)dx$

$=\int \cos xdx+\int \sin xdx$

$=\sin x-\cos x+C$

2. $\int 3\sin xdx$

$=3\times \int \sin xdx$

$=-3\cos x+C$

3. $\int 1+{\tan }^{2}xdx$

$=\int {\sec }^{2}xdx$

$=\tan x+C$

4. $3x^2/2-\tan x+C$

5. $-e^x+C$

6. $8e^x+C$

7. $\frac{1}{7}\ln \left|7x\right|+C$

8. with the substitution $x=a\tan \theta$, we have $dx=a{\sec }^2\theta d\theta$, and $x^2+a^2=a^2{\sec }^2\theta$, so that

$\int \frac{1}{x^2+a^2}dx$

$=\int \frac{a{\sec }^2\theta d\theta }{a^2{\sec }^2\theta }$

$=\int \frac{1}{a}d\theta$

$=\frac{1}{a}\theta +C$

$=\frac{1}{a}\arctan \frac{x}{a}+C$