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

The integral of 1/(2x^2+3x+5) is 2/(sqrt(31))arctan((4x+3)/(sqrt(31)))+C

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

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

\frac{2}{31} arctan (\frac{4 x + 3}{31}) + C

Solution steps

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

Complete the square

2 x^{2} + 3 x + 5 : 2 (x + \frac{3}{4})^{2} + \frac{31}{8}

= \int \frac{1}{2 ( x + \frac{3}{4} ) ^{2} + \frac{31}{8}} d x

Apply u-substitution

: \int \frac{8}{16 u ^{2} + 31} d u

= \int \frac{8}{16 u ^{2} + 31} d u

Take the constant out:

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

= 8 \cdot \int \frac{1}{16 u ^{2} + 31} d u

Apply Integral Substitution

= 8 \cdot \int \frac{1}{4 31 ( v ^{2} + 1 )} d v

Take the constant out:

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

= 8 \cdot \frac{1}{4 31} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 8 \cdot \frac{1}{4 31} arctan (v)

Substitute back

= 8 \cdot \frac{1}{4 31} arctan (\frac{4}{31} (x + \frac{3}{4}))

Simplify

8 \cdot \frac{1}{4 31} arctan (\frac{4}{31} (x + \frac{3}{4})) : \frac{2}{31} arctan (\frac{4 x + 3}{31})

= \frac{2}{31} arctan (\frac{4 x + 3}{31})

Add a constant to the solution

= \frac{2}{31} arctan (\frac{4 x + 3}{31}) + C

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What is the integral of 1/(2x^2+3x+5) ?
The integral of 1/(2x^2+3x+5) is 2/(sqrt(31))arctan((4x+3)/(sqrt(31)))+C