How to find indefinite integrals and antiderivatives

When we take a derivative, a constant disappears. An indefinite integral works back from the derivative, so we need a way to put that constant back. Let’s use the power rule and check what happens when we differentiate the result.

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Read the integral as a family of functions

If $F\prime \left(x\right)=f\left(x\right)$ on an interval, then every function $F\left(x\right)+C$ has derivative $f\left(x\right)$ there. That is what the indefinite integral represents.

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

Use the power rule for integration

For a power $x^a$ on an interval where it is defined, add one to the exponent and divide by the new exponent. This rule requires $a\ne -1$. For $a=-1$, the antiderivative is $\ln \left|x\right|+C$ on any interval that avoids zero.

$\begin{array}{l}\int x^{a}dx=\frac{x^{a+1}}{a+1}+C\\ \int 3x^{6}dx=\frac{3x^7}{7}+C\end{array}$

The derivative of $\frac{3x^7}{7}+C$ is $3x^6$, which checks the answer.

Integrate a polynomial one term at a time

$\int (2x^4-6x+2)dx$

$=\frac{2x^5}{5}-3x^2+2x+C$

Differentiate the result to get $2x^4-6x+2$. One constant at the end accounts for all the constants from the individual terms.

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