What is the integral of (6x+7)/((x+2)^2) ?

The integral of (6x+7)/((x+2)^2) is 6ln|x+2|+5/(x+2)+C

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integral of (6x+7)/((x+2)^2)

\int \frac{6 x + 7}{( x + 2 ) ^{2}} d x

6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + C

Solution steps

\int \frac{6 x + 7}{( x + 2 ) ^{2}} d x

Apply u-substitution

= \int \frac{6 u - 5}{u ^{2}} d u

Expand

\frac{6 u - 5}{u ^{2}} : \frac{6}{u} - \frac{5}{u ^{2}}

= \int \frac{6}{u} - \frac{5}{u ^{2}} d u

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{6}{u} d u - \int \frac{5}{u ^{2}} d u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

\int \frac{5}{u ^{2}} d u = - \frac{5}{u}

= 6 ln ∣ u ∣ - (- \frac{5}{u})

Substitute back

u = x + 2

= 6 ln ∣ x + 2 ∣ - (- \frac{5}{x + 2})

Simplify

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2}

Add a constant to the solution

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + C

\int \frac{1}{x ( ln ( x ) ) ^{3}} d x

y^{^{'}} = y (y + 1) x

\int \frac{1 + ln ( x )}{x ln ( x )} d x

\frac{d}{d x} (3 sin^{2} (x) cos (2 x))

\frac{\partial}{\partial x} ((\frac{x}{x - y})^{\frac{1}{3}})

What is the integral of (6x+7)/((x+2)^2) ?
The integral of (6x+7)/((x+2)^2) is 6ln|x+2|+5/(x+2)+C