What is the integral of x(x+a)(x+b) ?

The integral of x(x+a)(x+b) is (x^4)/4+(bx^3)/3+(ax^3)/3+(abx^2)/2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(x+a)(x+b)

\int x (x + a) (x + b) d x

\frac{x ^{4}}{4} + \frac{b x ^{3}}{3} + \frac{a x ^{3}}{3} + \frac{ab x ^{2}}{2} + C

Solution steps

\int x (x + a) (x + b) d x

Expand

x (x + a) (x + b) : x^{3} + b x^{2} + a x^{2} + ab x

= \int x^{3} + b x^{2} + a x^{2} + ab x 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 x^{3} d x + \int b x^{2} d x + \int a x^{2} d x + \int ab x d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int b x^{2} d x = \frac{b x ^{3}}{3}

\int a x^{2} d x = \frac{a x ^{3}}{3}

\int ab x d x = \frac{ab x ^{2}}{2}

= \frac{x ^{4}}{4} + \frac{b x ^{3}}{3} + \frac{a x ^{3}}{3} + \frac{ab x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{b x ^{3}}{3} + \frac{a x ^{3}}{3} + \frac{ab x ^{2}}{2} + C

y^{^{''}} + 20 y^{^{'}} + 500 = 12

\int \frac{1}{x - 1 + 1 + x} d x

y^{^{'}} = 2 + y - 2 x + 3

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

y

What is the integral of x(x+a)(x+b) ?
The integral of x(x+a)(x+b) is (x^4)/4+(bx^3)/3+(ax^3)/3+(abx^2)/2+C