Question

Evaluate the integral ∫(x² - 25x)dx using the indicated trigonometric substitution. (Use C for the constant of integration.)

Answer 1

The integral ∫(x² - 25x)dx is evaluated without trigonometric substitution as ⅓x³ - ⅔*25x² + C.

Step-by-step explanation:

The student has asked to evaluate the integral ∫(x² - 25x)dx using a trigonometric substitution, but the required substitution is not provided. Typically, trigonometric substitution is used for integrals involving the square root of a square minus a variable squared, or a variable squared minus a square, etc. In this case, however, the integral can be easily computed without trigonometric substitution.

Using basic rules of integration, we can integrate the polynomial term by term:

1. For the term x², the integral is ⅓x³.
2. For the term -25x, the integral is -⅒*25x² or -⅔*25x².

Combining these results, the integral becomes ⅓x³ - ⅔*25x² + C, where C is the constant of integration.