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

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

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

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

\frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3) + C

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

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

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

\int 3 x^{2} (3 + x^{2})^{\frac{1}{2}} d x = 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}})

= x (3 + x^{2})^{\frac{3}{2}} - 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}})

Simplify

x (3 + x^{2})^{\frac{3}{2}} - 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}}) : \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3)

= \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3)

Add a constant to the solution

= \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3) + C

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\frac{d y}{d t} + e^{t} y = - 4 e^{t}