2sinx-3cosx-3sinx-4cosx-dx-

Question Number 23796 by tapan das last updated on 06/Nov/17

\int \frac{2 sinx + 3 cosx}{3 sinx + 4 cosx} dx

Answered by ajfour last updated on 06/Nov/17

2 \sin x + 3 \cos x = A (3 \sin x + 4 \cos x)

+ B (3 \cos x - 4 \sin x)

\Rightarrow 3 A - 4 B = 2 a n d

4 A + 3 B = 3

\Rightarrow A = \frac{18}{25}, B = \frac{1}{25}; S o

\int \frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} d x = \frac{18}{25} \int d x +

+ \frac{1}{25} \int \frac{3 \cos x - 4 \sin x}{3 \sin x + 4 \cos x} d x

= \frac{18 x}{25} + \frac{1}{25} \ln ∣ 3 \sin x + 4 \cos x ∣ + C .

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