Question 14

Evaluate the indefinite integral
\[
\int 6 \sin ^{4} v \cos v d v=
\]

Answer

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Answer

Finally, the indefinite integral \(\int 6 \sin ^{4} v \cos v d v\) is \(\boxed{1.2\sin^{5}v + C}\) where C is the constant of integration.

Steps

Step 1 ：Let's evaluate the indefinite integral \(\int 6 \sin ^{4} v \cos v d v\).

Step 2 ：We can rewrite the integral as \(6/5 \times \int du\).

Step 3 ：Here, \(u = \sin(v)^{5}\) and \(du = 5 \sin(v)^{4} \cos(v) dv\).

Step 4 ：Substituting these values, the integral simplifies to \(1.2 \sin(v)^{5}\).

Step 5 ：Finally, the indefinite integral \(\int 6 \sin ^{4} v \cos v d v\) is \(\boxed{1.2\sin^{5}v + C}\) where C is the constant of integration.