What is the integral of (x^2)/(sqrt(x^2-4x+13)) ?

The integral of (x^2)/(sqrt(x^2-4x+13)) is

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integral of (x^2)/(sqrt(x^2-4x+13))

\int \frac{x ^{2}}{x ^{2} - 4 x + 13} d x

9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13) + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - 4 x + 13} d x

Complete the square

x^{2} - 4 x + 13 : (x - 2)^{2} + 9

= \int \frac{x ^{2}}{( x - 2 ) ^{2} + 9} d x

Apply u-substitution

: \int \frac{( u + 2 ) ^{2}}{u ^{2} + 9} d u

= \int \frac{( u + 2 ) ^{2}}{u ^{2} + 9} d u

Expand

\frac{( u + 2 ) ^{2}}{u ^{2} + 9} : \frac{u ^{2}}{u ^{2} + 9} + \frac{4 u}{u ^{2} + 9} + \frac{4}{u ^{2} + 9}

= \int \frac{u ^{2}}{u ^{2} + 9} + \frac{4 u}{u ^{2} + 9} + \frac{4}{u ^{2} + 9} 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

= \int \frac{u ^{2}}{u ^{2} + 9} d u + \int \frac{4 u}{u ^{2} + 9} d u + \int \frac{4}{u ^{2} + 9} d u

\int \frac{u ^{2}}{u ^{2} + 9} d u = 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2})

\int \frac{4 u}{u ^{2} + 9} d u = 4 u^{2} + 9

\int \frac{4}{u ^{2} + 9} d u = 4 ln (\frac{1}{3} u + 9 + u^{2})

= 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2}) + 4 u^{2} + 9 + 4 ln (\frac{1}{3} u + 9 + u^{2})

Substitute back

u = x - 2

= 9 - \frac{1}{2} ln \frac{x - 2 + 9 + ( x - 2 ) ^{2}}{3} + \frac{1}{18} (x - 2) 9 + (x - 2)^{2} + 4 (x - 2)^{2} + 9 + 4 ln (\frac{1}{3} x - 2 + 9 + (x - 2)^{2})

Simplify

9 - \frac{1}{2} ln \frac{x - 2 + 9 + ( x - 2 ) ^{2}}{3} + \frac{1}{18} (x - 2) 9 + (x - 2)^{2} + 4 (x - 2)^{2} + 9 + 4 ln (\frac{1}{3} x - 2 + 9 + (x - 2)^{2}) : 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13)

= 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13)

Add a constant to the solution

= 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13) + C

\int \frac{e ^{4 x}}{2} d x

x \to 0 lim (x \cdot ln (x))

x \to 3 lim (4^{f (x)})

\int_{1}^{2 x} \frac{y}{x} d x

\frac{d y}{d x}, x y = 2 y + x

What is the integral of (x^2)/(sqrt(x^2-4x+13)) ?
The integral of (x^2)/(sqrt(x^2-4x+13)) is