What is the integral of (3x+2)/(sqrt(x^2-4x+20)) ?

The integral of (3x+2)/(sqrt(x^2-4x+20)) is 3sqrt(x^2-4x+20)+8ln(1/4 |x-2+sqrt(x^2-4x+20)|)+C

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integral of (3x+2)/(sqrt(x^2-4x+20))

\int \frac{3 x + 2}{x ^{2} - 4 x + 20} d x

3 x^{2} - 4 x + 20 + 8 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20) + C

Solution steps

\int \frac{3 x + 2}{x ^{2} - 4 x + 20} d x

Expand

\frac{3 x + 2}{x ^{2} - 4 x + 20} : \frac{3 x}{x ^{2} - 4 x + 20} + \frac{2}{x ^{2} - 4 x + 20}

= \int \frac{3 x}{x ^{2} - 4 x + 20} + \frac{2}{x ^{2} - 4 x + 20} d x

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{3 x}{x ^{2} - 4 x + 20} d x + \int \frac{2}{x ^{2} - 4 x + 20} d x

\int \frac{3 x}{x ^{2} - 4 x + 20} d x = 3 (x^{2} - 4 x + 20 + 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20))

\int \frac{2}{x ^{2} - 4 x + 20} d x = 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

= 3 (x^{2} - 4 x + 20 + 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)) + 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

Simplify

3 (x^{2} - 4 x + 20 + 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)) + 2 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20) : 3 x^{2} - 4 x + 20 + 8 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

= 3 x^{2} - 4 x + 20 + 8 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

Add a constant to the solution

= 3 x^{2} - 4 x + 20 + 8 ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20) + C

e^{\frac{x}{2}}

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

\int \frac{4 ^{x}}{ln ( 4 )} d x

\int 4 x - 10 x e^{- 0.2 x} d x

\int (3 x^{8} - 7 x^{3} + 7) d x

What is the integral of (3x+2)/(sqrt(x^2-4x+20)) ?
The integral of (3x+2)/(sqrt(x^2-4x+20)) is 3sqrt(x^2-4x+20)+8ln(1/4 |x-2+sqrt(x^2-4x+20)|)+C