∫ 1 √( 1+4 x ^ 2 ) #160; dx =∫ 1 √( 4( 1+4 x ^ 2 4 #160; ) #160; ) #160; dx =∫[ 1 2 . 1 √( 1 4 + x ^ 2 ) #160; #160; ] d ...

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Integration by Substitution

Solve: ∫1√1+...

Question

Solve: $\int \frac{1}{\sqrt{1 + 4 x^{2}}} d x$

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Solution

$\int \frac{1}{\sqrt{1 + 4 x^{2}}} d x = \int \frac{1}{\sqrt{4 (\frac{1 + 4 x^{2}}{4})}} d x = \int ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{2} . \frac{1}{\sqrt{\frac{1}{4} + x^{2}}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d x = \frac{1}{2} \int \frac{1}{\sqrt{{(\frac{1}{2})}^{2} + x^{2}}} d x = \frac{1}{2} ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{\frac{1}{2}} {tan}^{- 1} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x}{\frac{1}{2}} ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ + C = {tan}^{- 1} 2 x + C$

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