Basic formulas and methods of calculating integrals

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

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

Solved examples

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

Find the indefinite integral of a function $\int(3x^2-2x+1)dx$

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

Find the indefinite integral of a function $\int(\frac{4}{x^5}+4\sin{x}-\pi)dx$

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

2019-03-03