What is the integral of (5x+7)/(4x^2+12x+10) ?

The integral of (5x+7)/(4x^2+12x+10) is 5/8 ln|4x^2+12x+10|-1/4 arctan(2x+3)+C

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integral of (5x+7)/(4x^2+12x+10)

\int \frac{5 x + 7}{4 x ^{2} + 12 x + 10} d x

\frac{5}{8} ln 4 x^{2} + 12 x + 10 - \frac{1}{4} arctan (2 x + 3) + C

Solution steps

\int \frac{5 x + 7}{4 x ^{2} + 12 x + 10} d x

Expand

\frac{5 x + 7}{4 x ^{2} + 12 x + 10} : \frac{5 x}{4 x ^{2} + 12 x + 10} + \frac{7}{4 x ^{2} + 12 x + 10}

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{5 x}{4 x ^{2} + 12 x + 10} d x + \int \frac{7}{4 x ^{2} + 12 x + 10} d x

\int \frac{5 x}{4 x ^{2} + 12 x + 10} d x = \frac{5}{8} (ln 4 x^{2} + 12 x + 10 - 6 arctan (2 x + 3))

\int \frac{7}{4 x ^{2} + 12 x + 10} d x = \frac{7}{2} arctan (2 x + 3)

= \frac{5}{8} (ln 4 x^{2} + 12 x + 10 - 6 arctan (2 x + 3)) + \frac{7}{2} arctan (2 x + 3)

Simplify

\frac{5}{8} (ln 4 x^{2} + 12 x + 10 - 6 arctan (2 x + 3)) + \frac{7}{2} arctan (2 x + 3) : \frac{5}{8} ln 4 x^{2} + 12 x + 10 - \frac{1}{4} arctan (2 x + 3)

= \frac{5}{8} ln 4 x^{2} + 12 x + 10 - \frac{1}{4} arctan (2 x + 3)

Add a constant to the solution

= \frac{5}{8} ln 4 x^{2} + 12 x + 10 - \frac{1}{4} arctan (2 x + 3) + C

f (t) = \frac{( e ^{x} + 5 x )}{5 x}

x^{2} + 2, 2 x + 6

\int 10 tan^{2} (x) d x

\int 14 x - 3 d x

- sin (π x) (π^{2})

What is the integral of (5x+7)/(4x^2+12x+10) ?
The integral of (5x+7)/(4x^2+12x+10) is 5/8 ln|4x^2+12x+10|-1/4 arctan(2x+3)+C