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The topic of the given section is Indefinite Integrals.

Functions with the Same Derivative Differ by a Constant.

If functions f and g are differentiable then $f\text{'}\left(x\right)=g\text{'}\left(x\right)$when $f\left(x\right)=g\left(x\right)+C$for some constant C.

Indefinite Integral.

An indefinite Integral is a family of functions that differ by a constant.

$\int f\left(x\right)dx=F\left(x\right)+C.$

where F is the antiderivative of fso,$F\text{'}=f.$

Integrals of Power Functions.

$$\begin{gathered} \left(i\right)\int x^{k}dx=\frac{1}{k+1}{x}^{k+1}+C,\mathrm{Where}k\ne -1 \\ \left(ii\right)\int \frac{1}{x}dx=\ln \left|x\right|+C \end{gathered}$$

Integrals of Exponential Functions.

$$\begin{gathered} \left(i\right)\int e^{kx}dx=\frac{1}{k}{e}^{kx}+C,\mathrm{Where}k\ne 0. \\ \left(ii\right)\int b^{x}dx=\frac{1}{\ln b}{b}^x+C,\mathrm{where}b>1. \end{gathered}$$

Integrals of Certain Trigonometric Expressions.

$$\begin{gathered} \left(i\right)\int \sin \left(x\right)dx=-\cos x+C \\ \left(ii\right)\int \cos xdx=\sin x+C \\ \left(iii\right)\int se{c}^{2}xdx=\tan x+C \\ \left(iv\right)\int cs{c}^{2}xdx=-\mathrm{co}tx+C \\ \left(v\right)\int secx\tan xdx=secx+C \\ \left(vi\right)\int cscxcotxdx=-cscx+C \end{gathered}$$

Constant Multiple and Sum Rules for Indefinite Integrals.

$$\begin{gathered} \left(i\right)\int kf\left(x\right)dx=k\int f\left(x\right)dx \\ \left(ii\right)\int \left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx \end{gathered}$$

Reversing the Product, Quotient, and Chain Rules.

$$\begin{gathered} \int \left[f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)\right]dx=f\left(x\right)g\left(x\right)+C \\ \int \frac{\left[f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)\right]}{{\left[g\left(x\right)\right]}^2}dx=\frac{f\left(x\right)}{g\left(x\right)}+C \\ \int \left[f\text{'}\left(g\right(x\left)\right)g\text{'}\left(x\right)\right]dx=f\left(g\right(x))+C \end{gathered}$$