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Question

$$\displaystyle\int \dfrac {\sin (2x)}{1 + \cos^{2}x} dx$$ is equal to


A
12log(1+cos2x)+c
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B
2log(1+cos2x)+c
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C
12log(1+cos2x)+c
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D
clog(1+cos2x)
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Solution

The correct option is B $$c - \log (1 + \cos^{2} x)$$
Let $$I = \displaystyle \int \dfrac {2\sin x \cos x}{1 +\cos^{2} x} dx$$
Put $$1 + \cos^{2} x = t$$
$$\Rightarrow -2\cos x \sin x dx = dt$$
$$\therefore I =\displaystyle  \int -\dfrac {dt}{t} = -\log t + c$$
$$= -\log (1 + \cos^{2} x) + c$$

Mathematics

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