Question

c) $\int (3t + 1)^{10} dt =$

d) $\int \cos^5 x \\ \sin x \\, dx =$

Explanation:

Part (c)

Step1: Use Substitution

Let \( u = 3t + 1 \), then \( du = 3dt \), so \( dt=\frac{du}{3} \).
The integral becomes \( \int u^{10}\cdot\frac{du}{3} \).

Step2: Integrate Using Power Rule

The power rule for integration is \( \int x^n dx=\frac{x^{n + 1}}{n+1}+C \) (\( n
eq - 1 \)).
So \( \frac{1}{3}\int u^{10}du=\frac{1}{3}\cdot\frac{u^{11}}{11}+C \).

Step3: Substitute Back \( u \)

Substitute \( u = 3t + 1 \) back into the expression: \( \frac{(3t + 1)^{11}}{33}+C \).

Part (d)

Step1: Use Substitution

Let \( u=\cos x \), then \( du=-\sin x dx \), so \( -\ du=\sin x dx \).
The integral becomes \( \int u^{5}\cdot(-du) \).

Step2: Integrate Using Power Rule

\( -\int u^{5}du=-\frac{u^{6}}{6}+C \).

Step3: Substitute Back \( u \)

Substitute \( u = \cos x \) back into the expression: \( -\frac{\cos^{6}x}{6}+C \).

Answer:

• For part (c): \(\boldsymbol{\frac{(3t + 1)^{11}}{33}+C}\)
• For part (d): \(\boldsymbol{-\frac{\cos^{6}x}{6}+C}\)