How to Integrate Exponents

By Karl Wallulis
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The general rule for integrating exponential terms consists of three steps: performing u-substitution, finding the antiderivative and then substituting the x values back into the equation. Rules that hold true for integrals and u-subsitution, such as moving coefficients outside the integral and eliminating all x terms when performing u-substitution, are also true when integrating exponential terms and are often essential in finding the integral.

Rewrite the integral in terms of u by substituting u for the exponential term. For example, if you are integrating the expression e^(x^4) x (8x^3), you would perform a u-substitution on the term x^4, yielding (e^u) x (8x^3).

Write an equation for du in terms of x and dx by finding the deriviative of u with respect to x. For example, if u is x^4, the derivative of x^4 is 4x^3, so du/dx = 4x^3, therefore du = 4x^3 dx.

Multiply or divide du by a constant so you can substitute the du term for the remaining x and dx terms in the integral. For example, you would have to multiply du by 2 to get 2 du = 8x^3 dx, allowing you to substitute 2 du into the expression for 8x^3 dx in the integrand (e^u) x (8x^3), making it entirely in terms of u: 2e^u du.

Move any coefficient outside the integral. In the example, the coefficient 2 must be moved outside the integral before integrating, making it 2 times the integral of e^u du.

Integrate the expression using the formula for the antiderivative of an exponential term. The antiderivative of b^k is b^k ln b. Notice that if the base is e, the antiderivative is simply e^k because the natural log of e is 1. In the above example, the integral of e^u du is simply e^u + C.

Substitute the value of x back into the expression and multiply by the removed coefficients. In the example, multiplying by 2 and substituting x gives the value of the integral: 2e^(x^4) + C.

About the Author

Karl Wallulis has been writing since 2010. He has written for the Guide to Online Schools website, covering academic and professional topics for young adults looking at higher-education opportunities. Wallulis holds a Bachelor of Arts in psychology from Whitman College.