# 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 Use a change of variables to evaluate the following integral.
$$\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx$$ 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.
$$I=\frac{\cos^{8}x}{8}+(\frac{-5}{6})\cos^{6}x+\frac{\sin^{2}x}{2}+C$$