Question

Evaluate the following indefinite integrals using a proper u-substitution. (a) ∫t(1−ᵗ²⁾⁵

Answer 1

Use u-substitution with u = 1 - t² for the indefinite integral ∫t(1−t²)⁵ dt, leading to ±(1/12)(1 - t²)⁶ + C.

Step-by-step explanation:

To evaluate the indefinite integral ∫t(1−t²)⁵ dt using a proper u-substitution, we first choose u = 1 - t², which implies that du = -2t dt. After rearranging, we get -½ du = t dt, which can be used to substitute the t dt part of the integral. Thus, the integral becomes ∫-½ u⁵ du, which is straightforward to integrate and results in ±(-½)u⁶/6 + C = ±(1/12)(1 - t²)⁶ + C, where C is the constant of integration.