Question

homework 2: 1.5 substitution
score: 6/35 answered: 3/9
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question 4
0/4
consider the indefinite integral $int sin^{27}(x) cos(x) dx$.
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

Explanation:

Step1: Choose substitution variable

Let $u = \sin(x)$

Step2: Compute derivative for $du$

Differentiate $u$: $du = \cos(x) dx$

Step3: Substitute into integral

Replace $\sin(x)$ with $u$ and $\cos(x)dx$ with $du$:
$\int u^{27} du$

Step4: Integrate with respect to $u$

Apply power rule: $\frac{u^{28}}{28} + C$

Step5: Substitute back to $x$

Replace $u$ with $\sin(x)$: $\frac{\sin^{28}(x)}{28} + C$

Answer:

a) $u = \sin(x)$
$du = \cos(x) dx$
c) $\int u^{27} du$
d) $\frac{\sin^{28}(x)}{28}$