Evaluate the following integrals: 2x ^4x+4 ^2x-2 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

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Answer

$\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
Let $\sin^2\text{x}=\text{t}$
$\Rightarrow2\sin\text{x}\cos\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin(2\text{x})\text{ dx}=\text{dt}$
Now, $\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}+4-4-2}}$
$=\int\frac{\text{dt}}{\sqrt{(\text{t}+2)^2-\big(\sqrt6\big)^2}}$
$=\log\Big|\text{t}+2+\sqrt{(\text{t}+2)^2-6}\Big|+\text{C}$
$=\log\Big|\text{t}+2+\sqrt{\text{t}^2+4\text{t}-2}\Big|+\text{C}$
$=\log\Big|\sin^2\text{x}+2\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}\Big|+\text{C}$

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