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Table of basic indefinite functions integrals

Formulas table of basic rational, exponential and trigonometric functions indefinite integrals.

$$\begin{array}{|c|c|} \hline \ \ \ \ \ f(x) \ \ \ \ \ & \ \ \ \ \ \int f(x) \ dx \ \ \ \ \ \\\hline a & ax+\text{C} \\\hline x & \frac{x^{2}}{2}+\text{C} \\\hline x^a & \frac{x^{a+1}}{a+1}+\text{C} \\\hline \frac{1}{x} & \ln |x|+\text{C} \\\hline {e^x} & {e^x}+\text{C} \\\hline {a^x} & \frac{a^x}{\ln a}+\text{C} \\\hline \sin x & -\cos x+\text{C} \\\hline \cos x & \sin x+\text{C} \\\hline \frac{1}{\cos^2 x} & \tan \ x+\text{C} \\\hline \frac{1}{\sin^2 x} & -\cot \ x+\text{C} \\\hline \end{array}$$

To see how this table works with real math task, we will make some simple examples for you.

Example No. 1
Find the indefinite integral of a function $\int(3x^2-2x+1)dx$.

$$\begin{align} \int(3x^2-2x+1)dx & =3\cdot \frac{x^3}{3}-2\cdot \frac{x^2}{2}+x+\text{C} \\ & = x^3-x^2+x+\text{C} \\ \end{align}$$

Example No. 2
Find the indefinite integral of a function $\int(\frac{4}{x^5}+4\sin{x}-\pi)dx$.

$$\begin{align} \int\Bigl(\frac{4}{x^5}+4\sin{x}-\pi\Bigr)dx & = \int(4x^{-5}+4\sin{x}-\pi)dx \\ & = 4\cdot\frac{x^{-4}}{-4}-4\cos{x}-\pi x+\text{C} \\ & = -x^{-4}-4\cos{x}-\pi x+\text{C} \end{align}$$

2019-03-03

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