What is the integral of (4x-10)/(3+(x+1)^2) ?

The integral of (4x-10)/(3+(x+1)^2) is 2ln|x^2+2x+4|-(14)/(sqrt(3))arctan(1/(sqrt(3))(x+1))+C

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integral of (4x-10)/(3+(x+1)^2)

\int \frac{4 x - 10}{3 + ( x + 1 ) ^{2}} d x

2 ln x^{2} + 2 x + 4 - \frac{14}{3} arctan (\frac{1}{3} (x + 1)) + C

Solution steps

\int \frac{4 x - 10}{3 + ( x + 1 ) ^{2}} d x

Expand

\frac{4 x - 10}{3 + ( x + 1 ) ^{2}} : \frac{4 x}{3 + ( x + 1 ) ^{2}} - \frac{10}{3 + ( x + 1 ) ^{2}}

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{4 x}{3 + ( x + 1 ) ^{2}} d x - \int \frac{10}{3 + ( x + 1 ) ^{2}} d x

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

\int \frac{10}{3 + ( x + 1 ) ^{2}} d x = \frac{10}{3} arctan (\frac{1}{3} (x + 1))

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

Simplify

4 (\frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))) - \frac{10}{3} arctan (\frac{1}{3} (x + 1)) : 2 ln x^{2} + 2 x + 4 - \frac{14}{3} arctan (\frac{1}{3} (x + 1))

= 2 ln x^{2} + 2 x + 4 - \frac{14}{3} arctan (\frac{1}{3} (x + 1))

Add a constant to the solution

= 2 ln x^{2} + 2 x + 4 - \frac{14}{3} arctan (\frac{1}{3} (x + 1)) + C

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What is the integral of (4x-10)/(3+(x+1)^2) ?
The integral of (4x-10)/(3+(x+1)^2) is 2ln|x^2+2x+4|-(14)/(sqrt(3))arctan(1/(sqrt(3))(x+1))+C