Question

homework 2: 1.5 substitution
score: 2/35 answered: 2/9
progress saved
question 3
0/4 pt
consider the indefinite integral $int x^{4}(x^{5}-2)^{34}dx$.
a) this can be transformed using the substitution
$u = \\\\$
which gives $du = \\\\$ (dont forget the differential $dx$ or $du$.)
c) performing the substitution in terms of $u$ gives the integral
$int \\\\$ .
d) evaluate the integral and simplify. your answer should be in terms of $x$, not $u$.
$\\\\$ $+c$
question help: video
submit question

Explanation:

Step1: Choose substitution variable

Let $u = x^5 - 2$

Step2: Compute derivative for $du$

Differentiate $u$: $du = 5x^4 dx$

Step3: Rewrite integral in terms of $u$

Rearrange $du$ to get $x^4 dx = \frac{1}{5}du$. Substitute into integral:
$$\int \frac{1}{5}u^{34} du$$

Step4: Integrate with respect to $u$

Use power rule $\int u^n du = \frac{u^{n+1}}{n+1}$:
$$\frac{1}{5} \cdot \frac{u^{35}}{35} = \frac{u^{35}}{175}$$

Step5: Substitute back $x$

Replace $u$ with $x^5 - 2$:
$$\frac{(x^5 - 2)^{35}}{175}$$

Answer:

a) $u = x^5 - 2$
$du = 5x^4 dx$
c) $\frac{1}{5}u^{34}$
d) $\frac{(x^5 - 2)^{35}}{175}$