Question

evaluate the integral.
∫ (9y⁸ dy)/(y⁹ - 14)
∫ (9y⁸ dy)/(y⁹ - 14) = □

Explanation:

Step1: Use substitution method

Let \( u = y^9 - 14 \), then find the derivative of \( u \) with respect to \( y \).
The derivative \( du/dy = 9y^8 \), so \( du = 9y^8 dy \).

Step2: Substitute into the integral

The integral \( \int \frac{9y^8 dy}{y^9 - 14} \) becomes \( \int \frac{du}{u} \) (since \( 9y^8 dy = du \) and \( u = y^9 - 14 \)).

Step3: Integrate \( \int \frac{du}{u} \)

The integral of \( \frac{1}{u} \) with respect to \( u \) is \( \ln|u| + C \), where \( C \) is the constant of integration.

Step4: Substitute back \( u = y^9 - 14 \)

We get \( \ln|y^9 - 14| + C \).

Answer:

\( \ln|y^9 - 14| + C \)