What is the integral of (a+bx)^2 ?

The integral of (a+bx)^2 is ((a+bx)^3)/(3b)+C

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integral of (a+bx)^2

\int (a + b x)^{2} d x

\frac{( a + b x ) ^{3}}{3 b} + C

Solution steps

\int (a + b x)^{2} d x

Apply u-substitution

: \int \frac{u ^{2}}{b} d u

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{b} \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{1}{b} \cdot \frac{u ^{3}}{3}

Substitute back

u = a + b x

= \frac{1}{b} \cdot \frac{( a + b x ) ^{3}}{3}

Simplify

\frac{1}{b} \cdot \frac{( a + b x ) ^{3}}{3} : \frac{( a + b x ) ^{3}}{3 b}

= \frac{( a + b x ) ^{3}}{3 b}

Add a constant to the solution

= \frac{( a + b x ) ^{3}}{3 b} + C

f (t) = 2 t^{3}

\frac{d y}{d x} = - (2 x + ln (y)) \cdot \frac{y}{x}

\frac{d y}{d x} = y^{2} x^{2}

f (x) = sin (x), at x = π

e^{x} sin (x), 0

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