What is the integral of 12(2x+7)(x+1)^2 ?

The integral of 12(2x+7)(x+1)^2 is 12((x^4)/2+(11x^3)/3+8x^2+7x)+C

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integral of 12(2x+7)(x+1)^2

\int 12 (2 x + 7) (x + 1)^{2} d x

12 (\frac{x ^{4}}{2} + \frac{11 x ^{3}}{3} + 8 x^{2} + 7 x) + C

Solution steps

\int 12 (2 x + 7) (x + 1)^{2} d x

Take the constant out:

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

= 12 \cdot \int (2 x + 7) (x + 1)^{2} d x

Expand

(2 x + 7) (x + 1)^{2} : 2 x^{3} + 11 x^{2} + 16 x + 7

= 12 \cdot \int 2 x^{3} + 11 x^{2} + 16 x + 7 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

= 12 (\int 2 x^{3} d x + \int 11 x^{2} d x + \int 16 x d x + \int 7 d x)

\int 2 x^{3} d x = \frac{x ^{4}}{2}

\int 11 x^{2} d x = \frac{11 x ^{3}}{3}

\int 16 x d x = 8 x^{2}

\int 7 d x = 7 x

= 12 (\frac{x ^{4}}{2} + \frac{11 x ^{3}}{3} + 8 x^{2} + 7 x)

Add a constant to the solution

= 12 (\frac{x ^{4}}{2} + \frac{11 x ^{3}}{3} + 8 x^{2} + 7 x) + C

f (x) = 2 e^{2 x}

x \to 1 - lim ((x - 1)^{2})

\int x^{2} (1 + 2 x^{3})^{5} d x

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

What is the integral of 12(2x+7)(x+1)^2 ?
The integral of 12(2x+7)(x+1)^2 is 12((x^4)/2+(11x^3)/3+8x^2+7x)+C