Evaluate the following integrals. \int\sin^3\theta\cos^{-2}\theta d\theta

amanf

amanf

Answered question

2021-10-05

Evaluate the following integrals.
sin3θcos2θdθ

Answer & Explanation

Theodore Schwartz

Theodore Schwartz

Skilled2021-10-06Added 99 answers

Let the given integral be I, then:
I=sin3θcos2θdθ
Rewrite the above equation, using trigonometric formula sin2θ+cos2θ=1
I=sin3θcos2θdθ
=(sin2θ)sinθcos2θdθ
=(1cosθ)sinθcos2θdθ (1)
Take:
t=cosθ (2)
Differentiate t w.r.t \theta:
dtdθ=ddθ(cosθ)
=sinθ
dt=sinθdθ (3)
Substitute the value of (2) and (3) in (1):
I=(1t2)t2dt
=(11t2)dt
=(1t2)dt
Solve the integral:
I=tt2+12+1+C
=t+1t+C
Where C is a constant.
Now, from (2) substitute the value of t in above equation:
I=cosθ+1cosθ+C

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