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

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

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

integral

(x^{2} + 1)^{\frac{1}{2}}

x (x^{2} + 1)^{\frac{1}{2}} - \frac{x ^{3}}{( x ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} - \frac{3}{4} ln \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + 1 - \frac{( 1 + x ^{2} ) ^{\frac{1}{2}}}{4 ( x + ( 1 + x ^{2} ) ^{\frac{1}{2}} )} + \frac{3}{4} ln \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} - 1 - \frac{( 1 + x ^{2} ) ^{\frac{1}{2}}}{4 ( x - ( 1 + x ^{2} ) ^{\frac{1}{2}} )} + 2 ln x + 1 + x^{2} - \frac{2 x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{1}{4 - x ^{2} - 4 x}

\frac{1}{ln ( y )}

- t^{2} + 2

(e^{- x})^{2} (x^{3})

x^{2} + y^{2}