What is the integral from-2 to 2x of (3t^2+2t) ?

The integral from-2 to 2x of (3t^2+2t) is 8x^3+4x^2+4

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integral from-2 to 2x of (3t^2+2t)

\int_{- 2}^{2 x} (3 t^{2} + 2 t) d t

8 x^{3} + 4 x^{2} + 4

Solution steps

\int_{- 2}^{2 x} 3 t^{2} + 2 t d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- 2}^{2 x} 3 t^{2} d t + \int_{- 2}^{2 x} 2 t d t

\int_{- 2}^{2 x} 3 t^{2} d t = 8 x^{3} + 8

\int_{- 2}^{2 x} 2 t d t = 2 (2 x^{2} - 2)

= 8 x^{3} + 8 + 2 (2 x^{2} - 2)

Simplify

8 x^{3} + 8 + 2 (2 x^{2} - 2) : 8 x^{3} + 4 x^{2} + 4

= 8 x^{3} + 4 x^{2} + 4

What is the integral from-2 to 2x of (3t^2+2t) ?
The integral from-2 to 2x of (3t^2+2t) is 8x^3+4x^2+4