Current Affairs JEE Main & Advanced

Integrals of the form \[\int_{{}}^{{}}{\frac{dx}{a+b\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x},\,\int_{{}}^{{}}{\frac{dx}{a+b\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x},}}\]\[\int_{{}}^{{}}{\frac{dx}{a\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+b\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}\mathbf{,}}\]\[\int_{{}}^{{}}{\frac{dx}{{{\mathbf{(}a\,\sin x+b\,\mathbf{cos}\,x\mathbf{)}}^{\mathbf{2}}}}\mathbf{,}}\int_{{}}^{{}}{\frac{dx}{a\,+b\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+c\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}}\]

To evaluate the above forms of integrals proceed as follows:

(1) Divide both the numerator and denominator by \[{{\cos }^{2}}x.\]

(2) Replace \[{{\sec }^{2}}x\]in the denominator, if any by \[(1+{{\tan }^{2}}x).\]

(3) Put \[\tan x=t\,\,\,\Rightarrow \,\,{{\sec }^{2}}xdx=dt.\]

(4) Now, evaluate the integral thus obtained, by the method discussed earlier.