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Evaluate the integral tan1(sin2x1+cos2x)dx
(a)x+c
(b)x22+c
(c)x2+c
(d)x24+c

Answer
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Hint: We are asked to find the integral of the given term. Since this is a complex one we need to first solve the stuff inside the bracket and we need to simplify that. In this case we will use several identities of double angle involving cosine and sine and then after simplifying the matter inside the bracket we will integrate the whole term. We should be aware of all the trigonometric identities.

Complete step-by-step solution:
We have tan1(sin2x1+cos2x)dx.
The term inside the bracket is:
sin2x1+cos2x
We can use the following identity in place of sin2x:
sin2x=2sinxcosx
And in place of cos2x, we can use the following formula:
cos2x=cos2xsin2x
Putting in the fraction and simplifying it, we have:
sin2x1+cos2x=2sinxcosx1+(cos2xsin2x)
Now, we use the following important identity:
cos2x+sin2x=1
Using the same we get:
cos2x=1sin2x
Putting this in the denominator:
sin2x1+cos2x=2sinxcosx2cos2x
So, we can say that:
sin2x1+cos2x=tanx
We obtained this using the formula:
sinxcosx=tanx
Putting this term inside the bracket we have:
tan1(tanx)dx=xdx
And we know that:
xdx=x22+c
Hence, option (b)x22+c is correct.

Note: If the formulae related to the double angle of sine and cosine are not known then this becomes a very difficult integral to operate because the terms inside bracket cannot be solved so easily. Moreover, you should be aware about the integrals of some basic algebraic terms such as the polynomial terms.