Question

evaluate the integral using the indicated trigonometric substitution. (use c for the constant of integration.) x3 x2 + 25 dx , x = 5 tan(theta)

Answer 1

The solution of the integral found by trigonometric substitution is I = 25 · [(1 / 2) · (x / 5)² + ㏑[5 / √(x² + 25)]] + C.

How to evaluate an integral by trigonometric substitution

In this question we have a rational expression whose integral can be found by transforming algebraic expressions into analogous trigonometric expression, that is:

I = ∫ [x³ / (x² + 25)] dx

If tan θ = x / 5 and dx = 5 · sec² θ, then the integral becames into:

I = 25 ∫ tan³ θ dθ

This integral can be found manually by using part integration or almost immediately by using integral tables. The solution to this integral is:

I = 25 · [tan² θ / 2 + ㏑ (cos θ)] + C

I = 25 · [(1 / 2) · (x / 5)² + ㏑[5 / √(x² + 25)]] + C

The solution of the integral found by trigonometric substitution is I = 25 · [(1 / 2) · (x / 5)² + ㏑[5 / √(x² + 25)]] + C.

To learn more on integrals: https://brainly.com/question/14502499

#SPJ1