Integrals of polynomials A common mistake when dealing with exponential expressions is treating the exponent on e the same way we treat exponents in polynomial expressions. Use substitution to evaluate the indefinite integral \(\displaystyle ∫3x^2e^{2x^3}\,dx.\) Solution. Here we choose to let \(u\) equal the expression in the exponent on \(e\). Indefinite integrals may or may not exist, but when they do, there are some general rules you can follow to simplify the integration procedure. Let \(u=2x^3\) and \(du=6x^2\,dx\). Indefinite integral. Indefinite integrals are antiderivative functions. We cannot use the power rule for the exponent on .This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. The power rule for integrals allows us to find the indefinite (and later the definite) integrals of a variety of functions like polynomials, functions involving roots, and even some rational functions. If you can write it with an exponents, you probably can apply the power rule. Example \(\PageIndex{3}\): Using Substitution with an Exponential Function. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. We cannot use the power rule for the exponent on e.This can be especially confusing when we have both exponentials and polynomials in the same expression, as in the previous checkpoint. Then, divide by that same value.

To apply the rule, simply take the exponent and add 1.

A common mistake when dealing with exponential expressions is treating the exponent on the same way we treat exponents in polynomial expressions. ∫ 1 ⁄ … Example \(\PageIndex{3}\): Using Substitution with an Exponential Function. Use substitution to evaluate the indefinite integral \(∫3x^2e^{2x^3}dx.\) Solution. Answer to: Use the exponential rule to find the indefinite integral. Indefinite Integral Rules Common Indefinite Integral Rules ∫m dx = mx + c, for any number m. ∫x n dx = 1 ⁄ n + 1 x x + 1 + c, if n ≠ –1.