What is the integral of (x^3)/(x^2+30x+225) ?

The integral of (x^3)/(x^2+30x+225) is 1/2 (x+15)^2-45(x+15)+675ln|x+15|+(3375)/(x+15)+C

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integral of (x^3)/(x^2+30x+225)

\int \frac{x ^{3}}{x ^{2} + 30 x + 225} d x

\frac{1}{2} (x + 15)^{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ + \frac{3375}{x + 15} + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 30 x + 225} d x

Complete the square

x^{2} + 30 x + 225 : (x + 15)^{2}

= \int \frac{x ^{3}}{( x + 15 ) ^{2}} d x

Apply u-substitution

: \int \frac{( u - 15 ) ^{3}}{u ^{2}} d u

= \int \frac{( u - 15 ) ^{3}}{u ^{2}} d u

Expand

\frac{( u - 15 ) ^{3}}{u ^{2}} : u - 45 + \frac{675}{u} - \frac{3375}{u ^{2}}

= \int u - 45 + \frac{675}{u} - \frac{3375}{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 u d u - \int 45 d u + \int \frac{675}{u} d u - \int \frac{3375}{u ^{2}} d u

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

\int 45 d u = 45 u

\int \frac{675}{u} d u = 675 ln ∣ u ∣

\int \frac{3375}{u ^{2}} d u = - \frac{3375}{u}

= \frac{u ^{2}}{2} - 45 u + 675 ln ∣ u ∣ - (- \frac{3375}{u})

Substitute back

u = x + 15

= \frac{( x + 15 ) ^{2}}{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ - (- \frac{3375}{x + 15})

Simplify

\frac{( x + 15 ) ^{2}}{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ - (- \frac{3375}{x + 15}) : \frac{1}{2} (x + 15)^{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ + \frac{3375}{x + 15}

= \frac{1}{2} (x + 15)^{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ + \frac{3375}{x + 15}

Add a constant to the solution

= \frac{1}{2} (x + 15)^{2} - 45 (x + 15) + 675 ln ∣ x + 15 ∣ + \frac{3375}{x + 15} + C

t \to 2 lim (\frac{2 t}{4 - t ^{2}})

\frac{d y}{d x} = - 3 + \frac{2}{1} \frac{d y}{d x}

f (x) = 15 x ln (4 x)

\frac{d}{d x} (\frac{7 - x e ^{x}}{x + e ^{x}})

\frac{( s - 1 )}{2 s ^{2} + 8 s + 11}

What is the integral of (x^3)/(x^2+30x+225) ?
The integral of (x^3)/(x^2+30x+225) is 1/2 (x+15)^2-45(x+15)+675ln|x+15|+(3375)/(x+15)+C