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Question

sin113(x).cos13(x)dx


A

3.cot23(x)73.cot83(x)5+c

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B

3.cot23(x)23.cot83(x)5+c

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C

3.cot23(x)23.cot83(x)8+c

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D

3.cot23(x)53.cot83(x)8+c

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Solution

The correct option is C

3.cot23(x)23.cot83(x)8+c


We can see that the given integral is also sinmx.cosnx dx . Where m & n are rational numbers. So if m+n22 is negative then we can either substitute cotx or tanx = t whichever is found suitable.

Here, m+n22=(113)+(13)22=3

So, we can either substitute cotx = t or tanx = t

sin113(x).cos13(x)dx

Let’s rewrite the expression to get either tanx or cotx in it

cos13(x)sin13(x).sin4(x)dxOr cot13(x)sin4(x)dxOr cot13(x)(cosec2x)2dxOr cot13(x)(1+cot2(x))(cosec2(x))dx

Let’s substitute cot (x) = t

cosec2x.dx=dtt13.(1+t2)dt=t13.dtt53.dtt2323t8383+c

Let’s substitute t = cot(x) in the above expression -

3.cot23(x)23.cot83(x)8+c


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