What is the integral of (4+x^2)^{3/2} ?

The integral of (4+x^2)^{3/2} is 3/2 x(x^2+4)^{1/2}+1/4 x(x^2+4)^{3/2}+6ln(1/2 |x+(x^2+4)^{1/2}|)+C

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integral of (4+x^2)^{3/2}

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

\frac{3}{2} x (x^{2} + 4)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 4)^{\frac{3}{2}} + 6 ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}}) + C

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

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

\int 3 x^{2} (4 + x^{2})^{\frac{1}{2}} d x = 48 - \frac{1}{32} x (4 + x^{2})^{\frac{1}{2}} + 4 ln \frac{x + ( 4 + x ^{2} ) ^{\frac{1}{2}}}{2} + \frac{1}{64} x (x^{2} + 4)^{\frac{3}{2}}

= x (4 + x^{2})^{\frac{3}{2}} - 48 - \frac{1}{32} x (4 + x^{2})^{\frac{1}{2}} + 4 ln \frac{x + ( 4 + x ^{2} ) ^{\frac{1}{2}}}{2} + \frac{1}{64} x (x^{2} + 4)^{\frac{3}{2}}

Simplify

x (4 + x^{2})^{\frac{3}{2}} - 48 - \frac{1}{32} x (4 + x^{2})^{\frac{1}{2}} + 4 ln \frac{x + ( 4 + x ^{2} ) ^{\frac{1}{2}}}{2} + \frac{1}{64} x (x^{2} + 4)^{\frac{3}{2}} : \frac{3}{2} x (x^{2} + 4)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 4)^{\frac{3}{2}} + 6 ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}})

= \frac{3}{2} x (x^{2} + 4)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 4)^{\frac{3}{2}} + 6 ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}})

Add a constant to the solution

= \frac{3}{2} x (x^{2} + 4)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 4)^{\frac{3}{2}} + 6 ln (\frac{1}{2} x + (x^{2} + 4)^{\frac{1}{2}}) + C

What is the integral of (4+x^2)^{3/2} ?
The integral of (4+x^2)^{3/2} is 3/2 x(x^2+4)^{1/2}+1/4 x(x^2+4)^{3/2}+6ln(1/2 |x+(x^2+4)^{1/2}|)+C