What is the integral of 1/((4+x^2)^{1/2)} ?

The integral of 1/((4+x^2)^{1/2)} is ln(1/2 |x+(x^2+4)^{1/2}|)+C

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integral of 1/((4+x^2)^{1/2)}

\int \frac{1}{( 4 + x ^{2} ) ^{\frac{1}{2}}} d x

ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}}) + C

Solution steps

\int \frac{1}{( 4 + x ^{2} ) ^{\frac{1}{2}}} d x

Apply Integration By Parts

: \frac{x}{( 4 + x ^{2} ) ^{\frac{1}{2}}} - \int - \frac{x ^{2}}{( 4 + x ^{2} ) ^{\frac{3}{2}}} d x

= \frac{x}{( 4 + x ^{2} ) ^{\frac{1}{2}}} - \int - \frac{x ^{2}}{( 4 + x ^{2} ) ^{\frac{3}{2}}} d x

\int - \frac{x ^{2}}{( 4 + x ^{2} ) ^{\frac{3}{2}}} d x = - ln (\frac{1}{2} x + (4 + x^{2})^{\frac{1}{2}}) + \frac{x}{( 4 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{( 4 + x ^{2} ) ^{\frac{1}{2}}} - (- ln (\frac{1}{2} x + (4 + x^{2})^{\frac{1}{2}}) + \frac{x}{( 4 + x ^{2} ) ^{\frac{1}{2}}})

Simplify

= ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}})

Add a constant to the solution

= ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}}) + C

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What is the integral of 1/((4+x^2)^{1/2)} ?
The integral of 1/((4+x^2)^{1/2)} is ln(1/2 |x+(x^2+4)^{1/2}|)+C