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webwork 5 - topics 10 -
(1 point)
evaluate the integral using an appropriate substitution.
$$ int \frac { x ^ { 7 } + 1 } { sqrt { x ^ { 8 } + 8 x } } d x = + c $$

Explanation:

Step1: Choose substitution

Let $u = x^8 + 8x$.

Step2: Compute derivative of u

$du = (8x^7 + 8)dx = 8(x^7 + 1)dx$, so $(x^7 + 1)dx = \frac{du}{8}$.

Step3: Rewrite integral in terms of u

$\int \frac{x^7 + 1}{\sqrt{x^8 + 8x}}dx = \int \frac{1}{\sqrt{u}} \cdot \frac{du}{8} = \frac{1}{8}\int u^{-1/2}du$.

Step4: Integrate using power rule

$\frac{1}{8} \cdot \frac{u^{1/2}}{1/2} + C = \frac{1}{4}u^{1/2} + C$.

Step5: Substitute back u

$\frac{1}{4}\sqrt{x^8 + 8x} + C$.

Answer:

$\frac{1}{4}\sqrt{x^8 + 8x}$