Question

exercises 1.5 substitution
score: 30/170 answered: 3/17
question 4
evaluate the integral
\\(\int x^5 (x^6 - 1)^{13} dx\\)
by making the substitution \\(u = x^6 - 1\\).
+ c
note: your answer should be in terms of \\(x\\) and not \\(u\\).
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Explanation:

Step1: Define substitution and find $du$

Let $u = x^6 - 1$. Differentiate $u$ with respect to $x$:
$$\frac{du}{dx} = 6x^5 \implies du = 6x^5 dx \implies x^5 dx = \frac{1}{6}du$$

Step2: Rewrite integral in terms of $u$

Substitute into the original integral:
$$\int x^5(x^6 - 1)^{13}dx = \int u^{13} \cdot \frac{1}{6}du = \frac{1}{6}\int u^{13}du$$

Step3: Integrate with respect to $u$

Use the power rule $\int u^n du = \frac{u^{n+1}}{n+1} + C$:
$$\frac{1}{6} \cdot \frac{u^{14}}{14} + C = \frac{u^{14}}{84} + C$$

Step4: Substitute back to $x$

Replace $u$ with $x^6 - 1$:
$$\frac{(x^6 - 1)^{14}}{84} + C$$

Answer:

$\frac{(x^6 - 1)^{14}}{84}$