Question

evaluate the indefinite integral.
\\(\int 7\sin^4 x \cos x dx\\)

Explanation:

Step1: Use substitution method

Let \( u = \sin x \), then \( du = \cos x dx \).
The integral becomes \( \int 7u^2 du \) (since \( \sin x = u \) and \( \cos x dx = du \), and the 7 is a constant factor).

Step2: Integrate the polynomial

Using the power rule for integration \( \int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\( n
eq - 1\)), for \( \int 7u^2 du \), we have:
\( 7\times\frac{u^{2 + 1}}{2+1}+C=\frac{7}{3}u^3 + C \)

Step3: Substitute back \( u=\sin x \)

We get \( \frac{7}{3}\sin^3 x + C \)

Answer:

\(\frac{7}{3}\sin^3 x + C\)