What is the integral of (2x+1)/((x^2+6x+10)(x+1)) ?

The integral of (2x+1)/((x^2+6x+10)(x+1)) is 12/5 arctan(x+3)-1/5 ln|x+1|+1/10 ln|x^2+6x+10|+C

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integral of (2x+1)/((x^2+6x+10)(x+1))

\int \frac{2 x + 1}{( x ^{2} + 6 x + 10 ) ( x + 1 )} d x

\frac{12}{5} arctan (x + 3) - \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{10} ln x^{2} + 6 x + 10 + C

Solution steps

\int \frac{2 x + 1}{( x ^{2} + 6 x + 10 ) ( x + 1 )} d x

Apply Integration By Parts

: (2 x + 1) (\frac{1}{10} (- ln x^{2} + 6 x + 10 - 4 arctan (x + 3)) + \frac{1}{5} ln ∣ x + 1 ∣) - \int 2 (\frac{1}{10} (- ln x^{2} + 6 x + 10 - 4 arctan (x + 3)) + \frac{1}{5} ln ∣ x + 1 ∣) d x

= (2 x + 1) (\frac{1}{10} (- ln x^{2} + 6 x + 10 - 4 arctan (x + 3)) + \frac{1}{5} ln ∣ x + 1 ∣) - \int 2 (\frac{1}{10} (- ln x^{2} + 6 x + 10 - 4 arctan (x + 3)) + \frac{1}{5} ln ∣ x + 1 ∣) d x

\int 2 (\frac{1}{10} (- ln x^{2} + 6 x + 10 - 4 arctan (x + 3)) + \frac{1}{5} ln ∣ x + 1 ∣) d x = 2 (- \frac{1}{10} x ln x^{2} + 6 x + 10 - \frac{2}{5} x arctan (x + 3) + \frac{1}{5} x ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x + 1 ∣ - \frac{1}{10} ln x^{2} + 6 x + 10 - \frac{7}{5} arctan (x + 3))

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

Simplify

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

= \frac{12}{5} arctan (x + 3) - \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{10} ln x^{2} + 6 x + 10

Add a constant to the solution

= \frac{12}{5} arctan (x + 3) - \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{10} ln x^{2} + 6 x + 10 + C

x \to - 6 lim (\frac{x ^{2} - 8}{6 - x})

\int_{- 2}^{2} (- x^{2} + 4)^{2} d x

x \to 0 lim (x cos (x))

f (x) = x^{2} + 2, (- 5, 27)

x \to - \infty lim ((1)^{x})

What is the integral of (2x+1)/((x^2+6x+10)(x+1)) ?
The integral of (2x+1)/((x^2+6x+10)(x+1)) is 12/5 arctan(x+3)-1/5 ln|x+1|+1/10 ln|x^2+6x+10|+C