Since this is really asking for the most general anti-derivative we just need to reuse the final answer from the first example. Calculus Examples. Show Solution. Split the single integral into multiple integrals. ∫ dy ∫ d y. Given a function f (x), the indefinite integral (or antiderivative) of f (x) is a function F (x) whose derivative is equal to f (x). Common Indefinite Integral Rules. Integrals. Step-by-Step Examples. Since is constant with respect to , move out of the integral. Example 1 Evaluate each of the following indefinite integrals. ∫ 3 4√x3 + 7 x5 + 1 6√x dx ∫ 3 x 3 4 + 7 x 5 + 1 6 x d x. ∫ (w+ 3√w)(4 −w2)dw ∫ ( w + w 3) ( 4 − w 2) d w. ∫ 4x10 −2x4+15x2 x3 … ∫sin x dx = −cos x + c. ∫cos x dx = sin x + c. ∫e x dx = e x + c.
∫m dx = mx + c, for any number m. ∫x n dx = 1 ⁄ n + 1 x x + 1 + c, if n ≠ –1. The indefinite integral is, ∫ x 4 + 3 x − 9 d x = 1 5 x 5 + 3 2 x 2 − 9 x + c ∫ x 4 + 3 x − 9 d x = 1 5 x 5 + 3 2 x 2 − 9 x + c. A couple of warnings are now in order. Evaluate the Integral. This means that F ' (x) = f (x). ∫ 1 ⁄ x dx = ln |x| + c, for x ≠ 0. Calculus.

∫ 5t3−10t−6 +4dt ∫ 5 t 3 − 10 t − 6 + 4 d t. ∫ x8 +x−8dx ∫ x 8 + x − 8 d x.
By the Power Rule, the integral of with respect to is .