Integrals Involving Exponential and Logarithmic Functions: Learn It 2
Integrals Involving Logarithmic Functions
Integrating functions of the form f(x)=x−1 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.
Use substitution. Let u=x4+3x2, then du=4x3+6x. Alter du by factoring out the 2.
Thus,
du=(4x3+6x)dx=2(2x3+3x)dx12du=(2x3+3x)dx.
Rewrite the integrand in u:
∫(2x3+3x)(x4+3x2)−1dx=12∫u−1du.
Then we have
12∫u−1du=12ln|u|+C=12ln|x4+3x2|+C.
Watch the following video to see the worked solution to this example.
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Find the antiderivative of the log function log2x.
Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have
∫log2xdx=xln2(lnx−1)+C.
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 ∫π/20sinx1+cosxdx.
We need substitution to evaluate this problem. Let u=1+cosx,, so du=−sinxdx. Rewrite the integral in terms of u, changing the limits of integration as well. Thus,