Evaluate the indefinite integral
\[
\int 8 \sin ^{6} x \cos x d x=
\]

Answer

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Answer

The indefinite integral is \(\boxed{\frac{8}{7} \sin^7 x + C}\)

Steps

Step 1 ：Given the integral \(\int 8 \sin ^{6} x \cos x d x\)

Step 2 ：Let's use the substitution method. We substitute \(u = \sin x\) and \(du = \cos x dx\)

Step 3 ：Rewriting the integral in terms of \(u\), we get \(\int 8u^6 du\)

Step 4 ：Evaluating this integral, we get \(\frac{8}{7}u^7 + C\), where \(C\) is the constant of integration

Step 5 ：Substituting back \(u = \sin x\), we get the final answer in terms of \(x\)

Step 6 ：The indefinite integral is \(\boxed{\frac{8}{7} \sin^7 x + C}\)