Question

evaluate the integral (int x^{7}(x^{8}-4)^{44}dx,) by making the appropriate substitution: (u =) (int x^{7}(x^{8}-4)^{44}dx =) note: your answer should be in terms of (x) and not (u). question help: video message instructor submit question jump to answer

Explanation:

Step1: Choose substitution variable

Let $u = x^8 - 4$

Step2: Compute derivative of $u$

$\frac{du}{dx} = 8x^7 \implies du = 8x^7 dx \implies x^7 dx = \frac{1}{8}du$

Step3: Rewrite integral in terms of $u$

$\int x^7(x^8 - 4)^{44}dx = \int u^{44} \cdot \frac{1}{8}du = \frac{1}{8}\int u^{44}du$

Step4: Integrate using power rule

$\frac{1}{8} \cdot \frac{u^{45}}{45} + C = \frac{u^{45}}{360} + C$

Step5: Substitute back to $x$

Replace $u$ with $x^8 - 4$

Answer:

$u = x^8 - 4$
$\frac{(x^8 - 4)^{45}}{360} + C$