What is the integral of (3x+7)/((x+4)^2) ?

The integral of (3x+7)/((x+4)^2) is 3ln|x+4|+5/(x+4)+C

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integral of (3x+7)/((x+4)^2)

\int \frac{3 x + 7}{( x + 4 ) ^{2}} d x

3 ln ∣ x + 4 ∣ + \frac{5}{x + 4} + C

Solution steps

\int \frac{3 x + 7}{( x + 4 ) ^{2}} d x

Apply u-substitution

: \int \frac{3 u - 5}{u ^{2}} d u

= \int \frac{3 u - 5}{u ^{2}} d u

Expand

\frac{3 u - 5}{u ^{2}} : \frac{3}{u} - \frac{5}{u ^{2}}

= \int \frac{3}{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{3}{u} d u - \int \frac{5}{u ^{2}} d u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

= 3 ln ∣ u ∣ - (- \frac{5}{u})

Substitute back

u = x + 4

= 3 ln ∣ x + 4 ∣ - (- \frac{5}{x + 4})

Simplify

= 3 ln ∣ x + 4 ∣ + \frac{5}{x + 4}

Add a constant to the solution

= 3 ln ∣ x + 4 ∣ + \frac{5}{x + 4} + C

x^{x^{2}}

\frac{d}{d y} (x e^{x y})

\int (x^{2} - 3 x + 6) d x

\int 2 x cos^{2} (x) d x

\frac{d}{d x} (x^{2} - 3 x - 4)

What is the integral of (3x+7)/((x+4)^2) ?
The integral of (3x+7)/((x+4)^2) is 3ln|x+4|+5/(x+4)+C