Explained: "Evaluate the following integrals. ∫sin20cos50d0"

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Answered question

2021-02-16

Evaluate the following integrals.

\int \sin^{2} 0 \cos^{5} 0 d 0

casincal

Skilled2021-02-17Added 82 answers

Step 1
Consider the provided integral,

\int \sin^{2} 0 \cos^{5} 0 d 0

Evaluate the following integrals.
Using the trigonometric identity,

\sin^{2} (0) \cos^{5} (0) d 0 = \int {(1 - \sin^{2} (0))}^{2} \cos (0) \sin^{2} (0) d 0

Apply u-Substitution method,

u = \sin 0 \Rightarrow d u = \cos 0 d 0

Step 2
Therefore,

\int \sin^{2} (0) \cos^{5} (0) d 0 = \int u^{2} {(1 - u^{2})}^{2} d u

= \int u^{2} - 2 u^{4} + u^{6} d u

= \int u^{2} d u - \int 2 u^{4} d u + \int u^{6} d u

= \frac{u^{3}}{3} - \frac{2 u^{5}}{5} + \frac{u^{7}}{7} + C

Substitute back,

\int \sin^{2} (0) \cos^{5} (0) d 0 = \frac{\sin^{3} 0}{3} - \frac{2 \sin^{5} 0}{5} + \frac{\sin^{7} 0}{7} + C

Hence.

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