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.