Question

Use a change of variables to evaluate the following integral. \int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx

Applications of integrals
ANSWERED
asked 2021-05-23
Use a change of variables to evaluate the following integral.
\(\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx\)

Answers (1)

2021-05-24

Step 1
The given integral is,
\(\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx\)
Step 2
The above integral can be split into the following integrals.
\(I=\int -\cos^{7}x\sin x dx+\int 5\cos^{5}x\sin x dx+\int \cos x \sin x dx\)
Step 3
The first integral is evaluated using the substitution as follows.
\(\int -\cos^{7}x\sin x dx=-\int -u^{7}du (\text{Use the substitution,} u=\cos x)\)
\(=\frac{u^{8}}{8}+c_{1}\)
\(=\frac{\cos^{8}x}{8}+c_{1}\)
Step 4
The second integral is evaluated using the substitution as follows.
\(\int 5\cos^{5}x\sin x dx = 5\int \cos^{5}x\sin x dx\)
\(=-5\int u^{5}du\) (Use the substitution, \(u = \cos x\))
\(=\frac{-5}{6}u^{6}+c_{2}\)
\(=\frac{-5}{6}\cos^{6}x+c_{2}\)
Step 5
The third integral is evaluated using the substitution as follows.
\(\int \cos x \sin x dx = \int u du\) (Use the substitution, \(u = \sin x\))
\(=\frac{u^{2}}{2}+c_{3}\)
\(=\frac{\sin^{2}x}{2}+c_{3}\)
Step 6
Thus, the integral \(\int\) can be written as follows.
\(I=\frac{\cos^{8}x}{8}+(\frac{-5}{6})\cos^{6}x+\frac{\sin^{2}x}{2}+C\)

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