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If \int \frac{dx}{\cos ^{3}x\sqrt{2\sin 2x}} = (tanx)^{A} + C(tanx)^{B} + k , 

where k is a constant of integration, then A + B + C equals :


  • Option 1)


  • Option 2)


  • Option 3)


  • Option 4)



Answers (1)


As learnt in concept

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.



- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )



 \int \frac{dx}{\cos ^{3}x\sqrt{2\sin x\:\cos x\times 2 }}

=\frac{1}{2}\int \frac{\sec ^{4}xdx}{\sqrt{\\tan x}}

=\frac{1}{2}\int \frac{\left ( 1+\tan ^{2}x \right )\sec ^{2}x \:dx}{\sqrt{\tan x}}

=\frac{1}{2}\int \left ( \tan x \right )^{\frac{-1}{2}} \sec ^{2}x\:dx+\frac{1}{2}\int \left ( \tan x \right )^{\frac{3}{2}} \sec ^{2}x \:dx

=\frac{1}{2}\frac{\left ( \tan x \right )^{\frac{1}{2}}}{\frac{1}{2}}+\frac{1}{2}\frac{\left ( \tan x \right )^{\frac{5}{2}}}{\frac{5}{2}}+C

A=\frac{1}{2}; B=\frac{5}{2}; C=\frac{1}{5}



Option 1)


Incorrect option    

Option 2)


Correct option

Option 3)


Incorrect option    

Option 4)


Incorrect option    

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