What is the integral of (x^2+3)/(xsqrt(x^2-4)) ?

The integral of (x^2+3)/(xsqrt(x^2-4)) is sqrt(x^2-4)+3/2 arctan(1/2 sqrt(x^2-4))+C

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integral of (x^2+3)/(xsqrt(x^2-4))

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u + 3}{2 u u - 4} d u

= \int \frac{u + 3}{2 u u - 4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{u + 3}{u u - 4} d u

Expand

\frac{u + 3}{u u - 4} : \frac{1}{u - 4} + \frac{3}{u u - 4}

= \frac{1}{2} \cdot \int \frac{1}{u - 4} + \frac{3}{u u - 4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{u - 4} d u = 2 u - 4

\int \frac{3}{u u - 4} d u = 3 arctan (\frac{1}{2} u - 4)

= \frac{1}{2} (2 u - 4 + 3 arctan (\frac{1}{2} u - 4))

Substitute back

u = x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x^2+3)/(xsqrt(x^2-4)) ?
The integral of (x^2+3)/(xsqrt(x^2-4)) is sqrt(x^2-4)+3/2 arctan(1/2 sqrt(x^2-4))+C