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Question

If (1+sin x)f(x)dx=23 (1+sin x)3/2+C,then f(x) is
.

A
cos x
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B
sin x
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C
tan x
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D
1
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Solution

The correct option is A cos x
Here, the important thing to remember is that integration is the reverse process of differentiation i.e.
If
g(x)dx=F(x)+C
, then
dF(x)dx=g(x).
Here, letg(x)dx=(1+sinx)f(x) dxg(x)=(1+sinx)f(x)also F(x)=23 (1+sinx)3/2+Cg(x)=F(x)g(x)=dF(x)dx

Thus, substituting the function values in the above equation we get,ddx(23 (1+sinx)3/2+C)=f(x)1+sinx 23.32 (1+sinx)1/2cos x=f(x)1+sinx (1+sinx)1/2cos x=f(x)1+sinx cos x=f(x)

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