What is the integral from 4 to 1 of 4x+3x^2-2x^3 ?

The integral from 4 to 1 of 4x+3x^2-2x^3 is 69/2

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integral from 4 to 1 of 4x+3x^2-2x^3

\int_{4}^{1} 4 x + 3 x^{2} - 2 x^{3} d x

\frac{69}{2}

Decimal Notation

34.5

Solution steps

\int_{4}^{1} 4 x + 3 x^{2} - 2 x^{3} d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{1}^{4} 4 x + 3 x^{2} - 2 x^{3} 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_{1}^{4} 4 x d x + \int_{1}^{4} 3 x^{2} d x - \int_{1}^{4} 2 x^{3} d x)

\int_{1}^{4} 4 x d x = 30

\int_{1}^{4} 3 x^{2} d x = 63

\int_{1}^{4} 2 x^{3} d x = \frac{255}{2}

= - (30 + 63 - \frac{255}{2})

Simplify

- (30 + 63 - \frac{255}{2}) : \frac{69}{2}

= \frac{69}{2}

\int_{- a}^{a} (a^{2} - t^{2}) d t

\int_{0}^{4} (2 t + 1) d t

\int_{0}^{20} x + sin (x) d x

\int_{1}^{4} 4 x - x^{2} d x

\int_{16}^{2 y + 4} 1

What is the integral from 4 to 1 of 4x+3x^2-2x^3 ?
The integral from 4 to 1 of 4x+3x^2-2x^3 is 69/2