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

The integral of (2x+3)/((x+7)^3) is -2/(x+7)+7/((x+7)^2)-3/(2(x+7)^2)+C

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

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

- \frac{2}{x + 7} + \frac{7}{( x + 7 ) ^{2}} - \frac{3}{2 ( x + 7 ) ^{2}} + C

Solution steps

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

Expand

\frac{2 x + 3}{( x + 7 ) ^{3}} : \frac{2 x}{( x + 7 ) ^{3}} + \frac{3}{( x + 7 ) ^{3}}

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

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

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

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

= - \frac{2}{x + 7} + \frac{7}{( x + 7 ) ^{2}} - \frac{3}{2 ( x + 7 ) ^{2}}

Add a constant to the solution

= - \frac{2}{x + 7} + \frac{7}{( x + 7 ) ^{2}} - \frac{3}{2 ( x + 7 ) ^{2}} + C

\frac{160 ( x + 1 )}{( x + 2 ) ^{2}}

y^{^{''}} - 16 y = 0

\int \frac{2 x ^{2}}{2 x ^{2} + 1} d x

\int x (2 x^{2} - 1)^{4} d x

\frac{d}{d x} (2 x + y)

What is the integral of (2x+3)/((x+7)^3) ?
The integral of (2x+3)/((x+7)^3) is -2/(x+7)+7/((x+7)^2)-3/(2(x+7)^2)+C