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Integration of Polynomials[edit]

Evaluate the following:

1. \int (x^2-2)^{2}\, dx


2. \int 8x^3\, dx


3. \int (4x^2+11x^3)\, dx


4. \int (31x^{32}+4x^3-9x^4) \,dx


5. \int 5x^{-2}\, dx


Indefinite Integration[edit]

Find the general antiderivative of the following:

6. \int (\cos x+\sin x)\, dx

\int (\cos x+\sin x)\, dx=\mathbf{\sin x-\cos x+C}

7. \int 3\sin x\, dx

\int 3\sin x\, dx=\mathbf{-3\cos(x)+C}

8. \int (1+\tan^2 x)\, dx

\begin{align}\int(1+\tan^{2}x)dx&=\int\sec^{2}x dx\\
&=\mathbf{\tan x+C}\end{align}

9. \int (3x-\sec^2 x)\, dx

\int (3x-\sec^2 x)\, dx=\mathbf{\frac{3x^{2}}{2}-\tan x+C}

10. \int -e^x\, dx

\int -e^x\, dx=\mathbf{-e^{x}+C}

11. \int 8e^x\, dx

\int 8e^x\, dx=\mathbf{8e^{x}+C}

12. \int \frac1{7x}\, dx

\int \frac1{7x}\, dx=\mathbf{\frac{1}{7}\ln|x|+C}

13. \int \frac1{x^2+a^2}\, dx


x=a\tan\theta;\qquad dx=a\sec^{2}\theta d\theta


\begin{align}\int\frac{1}{x^{2}+a^{2}}dx&=\int\frac{a\sec^{2}\theta d\theta}{a^{2}(\tan^{2}\theta+1)}\\
&=\int\frac{\sec^{2}\theta d\theta}{a\sec^{2}\theta}\\
&=\frac{1}{a}\int d\theta\\

Integration by parts[edit]

14. Consider the integral \int \sin(x) \cos(x)\,dx. Find the integral in two different ways. (a) Integrate by parts with u=\sin(x) and  v' =\cos(x). (b) Integrate by parts with u=\cos(x) and  v' =\sin(x). Compare your answers. Are they the same?


u=\sin x;\qquad du=\cos x dx
v=\sin x;\qquad dv=\cos x dx
\int\sin x\cos x dx=\sin^{2}x-\int\sin x\cos x dx & \implies & 2\int\sin x\cos x dx=\sin^{2}x\\
 & \implies & \int\sin x\cos x dx=\mathbf{\frac{\sin^{2}x}{2}}


u=\cos x;\qquad du=-\sin x dx
v=-\cos x\qquad dv=\sin x dx
\int\sin x\cos x dx=-\cos^{2}x-\int\sin x\cos x dx & \implies & 2\int\sin x\cos x dx=-\cos^{2}x\\
 & \implies & \int\sin x\cos x dx=\mathbf{-\frac{\cos^{2}x}{2}}

Notice that the answers in parts (a) and (b) are not equal. However, since indefinite integrals include a constant term, we expect that the answers we found will differ by a constant. Indeed