Compute {2 x}{² x+ x-6} \, dx

Correct Answer is : 252/552 [ln(cosx + 3) − ln(2 − cosx)] + C

2[ln(cosx + 3) + ln(2 − cosx)] + C

2/5 [ln(cosx + 3) + ln(2 − cosx)] + C

2[ln(cosx + 3) − ln(cosx − 2)] + C

2/5 [ln(cosx + 3) − ln(2 − cosx)] + C

Correct Answer is : 252/552 [ln(cosx + 3) − ln(2 − cosx)] + C

Test Index

UPSC NDA Math Model Paper 7

Question : 84 of 120

Marks: +1, -0

Compute

\int\limits \frac{2 \sin x}{\cos^{2} x+\cos x-6} \, dx

Solution:

I=\int\limits \frac{2 \sin x}{\cos^{2} x+\cos x-6} \, dx

\Rightarrow I=-2\int\limits \frac{-\sin x}{\cos^{2} x+\cos x-6} \, dx

By substitution let

\cos x=t \Rightarrow (-\sin x) \, dx = dt

\Rightarrow I=-2\int\limits \frac{1}{t^{2}+t-6} \, dt

\Rightarrow I=-2\int\limits \frac{1}{(t-2)(t+3)} \, dt

\Rightarrow I=-2\int\limits \frac{1}{5}\left[\frac{1}{t-2}-\frac{1}{t+3}\right] \, dt

\Rightarrow I=\frac{2}{5}\int\limits \left[\frac{1}{t+3}-\frac{1}{t-2}\right] \, dt

\Rightarrow I=\frac{2}{5}\left[\ln |t+3|-\ln |t-2|\right]+C

\Rightarrow I=\frac{2}{5}\left[\ln |\cos x+3|-\ln |\cos x-2|\right]+C

\because -1 \leq \cos x \leq 1

\Rightarrow I=\frac{2}{5}\left[\ln (\cos x+3)-\ln (2-\cos x)\right]+C

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