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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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