Find $\int \frac{(3\sin\theta-2)\cos\theta}{5-\cos^{2}\theta-4\sin\theta}\,d\theta$

$3\log|2-\sin\theta|+\dfrac{4}{2-\sin\theta}+C$

CBSE Class 12 · Indefinite IntegralsLong Answer

Find

\int \frac{( 3 s i n θ - 2 ) c o s θ}{5 - c o s ^{2} θ - 4 s i n θ} d θ

Answer: $3 lo g ∣2 - sin θ ∣ + \frac{4}{2 - sin θ} + C$

Simplify the denominator:

5 - cos^{2} θ - 4 sin θ = 5 - (1 - sin^{2} θ) - 4 sin θ = sin^{2} θ - 4 sin θ + 4 = (2 - sin θ)^{2}

. Put

t = sin θ

d t = cos θ d θ

\int \frac{3 t - 2}{( 2 - t ) ^{2}} d t .

Write

3 t - 2 = - 3 (2 - t) + 4

, so the integrand is

\frac{- 3}{2 - t} + \frac{4}{( 2 - t ) ^{2}}

\int [\frac{- 3}{2 - t} + \frac{4}{( 2 - t ) ^{2}}] d t = 3 lo g ∣2 - t ∣ + \frac{4}{2 - t} + C .

Restoring

t = sin θ

3 lo g ∣2 - sin θ ∣ + \frac{4}{2 - sin θ} + C

CBSE Class 12 Board · Subjective