Need Help?

  • Notes
  • Comments & Questions

Solve \(\int \sin ^{5} x \cos ^{2} x d x\)

(1) 
\(\int \sin ^{5} x \cos ^{2} x d x=\int \sin ^{4} x \cos ^{2} x \sin x d x \)


\(=\int\left(\sin ^{2} x\right)^{2} \cos ^{2} x \sin x d x\)
 

(2) 
\(\int\left(\sin ^{2} x\right)^{2} \cos ^{2} x \sin x d x=\int\left(1-\cos ^{2} x\right)^{2} \cos ^{2} x \sin x d x \)


\(=\int\left(1-2 \cos ^{2} x+\cos ^{4} x\right) \cos ^{2} x \sin x d x \)


\(=\int\left(\cos ^{2} x-2 \cos ^{4} x+\cos ^{6} x\right) \sin x d x \)

(3) let \(t=\cos x \longrightarrow d t=-\sin x d x\)


\(=-\int\left(\cos ^{2} x-2 \cos ^{4} x+\cos ^{6} x\right)(-\sin x) d x \)


\(\int \sin ^{5} x \cos ^{2} x d x=-\int\left(t^{2}-2 t^{4}+t^{6}\right) d t \)


\(=-\left[\frac{t^{3}}{3}-\frac{2 t^{5}}{5}+\frac{t^{7}}{7}\right]+c \)


\(\int \sin ^{5} x \cos ^{2} x d x=\frac{-\cos ^{3} x}{3}-\frac{2 \cos x^{5}}{5}+\frac{\cos ^{7} x}{7}+c \)

No comments yet

Join the conversation

Join Notatee Today!