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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)

    \frac{21}{5}

  • Option 2)

    \frac{16}{5}

  • Option 3)

    \frac{7}{10}

  • Option 4)

    \frac{27}{10}

 

Answers (1)

best_answer

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}

A+B+C=3+\frac{1}{5}

=\frac{16}{5}


Option 1)

\frac{21}{5}

Incorrect option    

Option 2)

\frac{16}{5}

Correct option

Option 3)

\frac{7}{10}

Incorrect option    

Option 4)

\frac{27}{10}

Incorrect option    

Posted by

Aadil

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