7. $int|2x^{3}+x^{2}-5|dx$ 8. $int e^{x}dx$ 9. $intsin3x dx$ 10. $int\frac{cos^{2}x}{1 - sin x}dx$ 11…

Explanation:

Step1: Integrate $\int(2x^{3}+x^{2}-5)dx$

Use power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$. $\int(2x^{3}+x^{2}-5)dx=2\int x^{3}dx+\int x^{2}dx-5\int dx=2\times\frac{x^{4}}{4}+\frac{x^{3}}{3}-5x+C=\frac{1}{2}x^{4}+\frac{1}{3}x^{3}-5x+C$

Step2: Integrate $\int e^{x}dx$

The antiderivative of $e^{x}$ is itself. $\int e^{x}dx=e^{x}+C$

Step3: Integrate $\int\sin(3x)dx$

Use substitution $u = 3x$, $du=3dx$. Then $\int\sin(3x)dx=\frac{1}{3}\int\sin(u)du=-\frac{1}{3}\cos(u)+C=-\frac{1}{3}\cos(3x)+C$

Step4: Integrate $\int\frac{\cos^{2}x}{1 - \sin x}dx$

Use $\cos^{2}x=1-\sin^{2}x=(1 + \sin x)(1 - \sin x)$. Then $\int\frac{\cos^{2}x}{1 - \sin x}dx=\int(1+\sin x)dx=\int dx+\int\sin xdx=x-\cos x+C$

Step5: Integrate $\int\sec^{3}x dx$

Use the reduction formula $\int\sec^{n}x dx=\frac{\sec^{n - 2}x\tan x}{n - 1}+\frac{n - 2}{n - 1}\int\sec^{n - 2}x dx$ for $n = 3$. $\int\sec^{3}x dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\int\sec x dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C$

Step6: Integrate $\int xe^{x}dx$

Use integration by parts with $u = x$, $dv=e^{x}dx$, $du = dx$, $v = e^{x}$. Then $\int xe^{x}dx=xe^{x}-\int e^{x}dx=xe^{x}-e^{x}+C=(x - 1)e^{x}+C$

Answer:

1. $\int(2x^{3}+x^{2}-5)dx=\frac{1}{2}x^{4}+\frac{1}{3}x^{3}-5x+C$
2. $\int e^{x}dx=e^{x}+C$
3. $\int\sin(3x)dx=-\frac{1}{3}\cos(3x)+C$
4. $\int\frac{\cos^{2}x}{1 - \sin x}dx=x-\cos x+C$
5. $\int\sec^{3}x dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C$
6. $\int xe^{x}dx=(x - 1)e^{x}+C$