What is the integral of (3x+1)/(x^2+5x+10) ?

The integral of (3x+1)/(x^2+5x+10) is

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integral of (3x+1)/(x^2+5x+10)

\int \frac{3 x + 1}{x ^{2} + 5 x + 10} d x

4 (3 (\frac{1}{8} ln 4 x^{2} + 20 x + 40 - \frac{5}{4 3} arctan (\frac{2}{15} x + \frac{5}{3})) + \frac{1}{2 15} arctan (\frac{2}{15} x + \frac{5}{3})) + C

Solution steps

\int \frac{3 x + 1}{x ^{2} + 5 x + 10} d x

Complete the square

x^{2} + 5 x + 10 : (x + \frac{5}{2})^{2} + \frac{15}{4}

= \int \frac{3 x + 1}{( x + \frac{5}{2} ) ^{2} + \frac{15}{4}} d x

Apply u-substitution

= \int \frac{4 ( 3 ( u - \frac{5}{2} ) + 1 )}{4 u ^{2} + 15} d u

Take the constant out:

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

= 4 \cdot \int \frac{3 ( u - \frac{5}{2} ) + 1}{4 u ^{2} + 15} d u

Expand

\frac{3 ( u - \frac{5}{2} ) + 1}{4 u ^{2} + 15} : \frac{3 ( u - \frac{5}{2} )}{4 u ^{2} + 15} + \frac{1}{4 u ^{2} + 15}

Apply the Sum Rule:

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

= 4 (\int \frac{3 ( u - \frac{5}{2} )}{4 u ^{2} + 15} d u + \int \frac{1}{4 u ^{2} + 15} d u)

\int \frac{3 ( u - \frac{5}{2} )}{4 u ^{2} + 15} d u = 3 (\frac{1}{8} ln 4 u^{2} + 15 - \frac{5}{4 3} arctan (\frac{2}{15} u))

\int \frac{1}{4 u ^{2} + 15} d u = \frac{1}{2 15} arctan (\frac{2}{15} u)

= 4 (3 (\frac{1}{8} ln 4 u^{2} + 15 - \frac{5}{4 3} arctan (\frac{2}{15} u)) + \frac{1}{2 15} arctan (\frac{2}{15} u))

Substitute back

u = x + \frac{5}{2}

= 4 (3 (\frac{1}{8} ln 4 (x + \frac{5}{2})^{2} + 15 - \frac{5}{4 3} arctan (\frac{2}{15} (x + \frac{5}{2}))) + \frac{1}{2 15} arctan (\frac{2}{15} (x + \frac{5}{2})))

Simplify

4 (3 (\frac{1}{8} ln 4 (x + \frac{5}{2})^{2} + 15 - \frac{5}{4 3} arctan (\frac{2}{15} (x + \frac{5}{2}))) + \frac{1}{2 15} arctan (\frac{2}{15} (x + \frac{5}{2}))) : 4 (3 (\frac{1}{8} ln 4 x^{2} + 20 x + 40 - \frac{5}{4 3} arctan (\frac{2}{15} x + \frac{5}{3})) + \frac{1}{2 15} arctan (\frac{2}{15} x + \frac{5}{3}))

= 4 (3 (\frac{1}{8} ln 4 x^{2} + 20 x + 40 - \frac{5}{4 3} arctan (\frac{2}{15} x + \frac{5}{3})) + \frac{1}{2 15} arctan (\frac{2}{15} x + \frac{5}{3}))

Add a constant to the solution

= 4 (3 (\frac{1}{8} ln 4 x^{2} + 20 x + 40 - \frac{5}{4 3} arctan (\frac{2}{15} x + \frac{5}{3})) + \frac{1}{2 15} arctan (\frac{2}{15} x + \frac{5}{3})) + C

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What is the integral of (3x+1)/(x^2+5x+10) ?
The integral of (3x+1)/(x^2+5x+10) is