Integrals Involving Exponential and Logarithmic Functions: Learn It 2

Integrals Involving Logarithmic Functions

Integrating functions of the form f(x)=x1 result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as f(x)=lnx and f(x)=logax, are also included in the rule.

integration formulas involving logarithmic functions

The following formulas can be used to evaluate integrals involving logarithmic functions.

x1dx=ln|x|+Clnxdx=xlnxx+C=x(lnx1)+Clogaxdx=xlna(lnx1)+C

Find the antiderivative of the function 3x10.

Find the antiderivative of 2x3+3xx4+3x2.

Find the antiderivative of the log function log2x.

The example below is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.

Find the definite integral of 0π/2sinx1+cosxdx.