Best solutions to - "Calculate the integrals. ∫sin3cos23xdx"

Tyra

Answered question

2021-10-15

Calculate the integrals.

\int \sin^{3} \cos^{\frac{2}{3}} x d x

Sally Cresswell

Skilled2021-10-16Added 91 answers

To integrate

\int \sin^{3} \cos^{\frac{2}{3}} x d x

\sin^{3} (x) \cos^{\frac{2}{3}} (x) d x = \int \sin^{2} (x) \sin (x) \cos^{\frac{2}{3}} (x) d x

\int (1 - \cos^{2} (z)) \sin (x) \cos^{\frac{2}{3}} (x) d x

Apply u-substitution:

u = \cos (x)

d u = - \sin x d x

= \int - u^{\frac{2}{3}} (1 - u^{2}) d u

= \int (- u^{\frac{2}{3} + u^{\frac{8}{3}}} d u

= - \int (u^{\frac{2}{3}}) d u + \int (+ u^{\frac{8}{3}}) d u

= \frac{3}{5} u^{\frac{5}{3}} + \frac{3}{11} u^{\frac{11}{3}} + C

Substitute back

u = \cos (x)

= - \frac{3}{5} \cos^{\frac{5}{3}} (x) + \frac{3}{11} \cos^{\frac{11}{3}} (x) + C

Hence

\int \sin^{3} \cos^{\frac{2}{3}} x d x = - \frac{3}{5} \cos^{\frac{5}{3} (x) + \frac{3}{11} \cos^{\frac{11}{3}} (x) + C}

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