Integrate x^3-2x from -1 to 2

\int_{-1}^{2}\left(x^3-2x\right)dx

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Answer

$\frac{3}{4}$

Step by step solution

Problem

$\int_{-1}^{2}\left(x^3-2x\right)dx$
1

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int_{-1}^{2}-2xdx+\int_{-1}^{2} x^3dx$
2

Taking the constant out of the integral

$\int_{-1}^{2} x^3dx-2\int_{-1}^{2} xdx$
3

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\left[\frac{x^{4}}{4}\right]_{-1}^{2}-2\int_{-1}^{2} xdx$
4

Evaluate the definite integral

$-2\int_{-1}^{2} xdx+\frac{{\left(-1\right)}^{4}}{4}\left(-1\right)+\frac{2^{4}}{4}$
5

Calculate the power

$-2\int_{-1}^{2} xdx+\frac{1}{4}\left(-1\right)+\frac{16}{4}$
6

Divide $16$ by $4$

$-2\int_{-1}^{2} xdx+\frac{1}{4}\left(-1\right)+4$
7

Multiply $-1$ times $\frac{1}{4}$

$-2\int_{-1}^{2} xdx-\frac{1}{4}+4$
8

Subtract the values $4$ and $-\frac{1}{4}$

$\frac{15}{4}-2\int_{-1}^{2} xdx$
9

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{15}{4}-2\left[\frac{1}{2}x^2\right]_{-1}^{2}$
10

Evaluate the definite integral

$\left(2^2\cdot 0.5-1\cdot {\left(-1\right)}^2\cdot 0.5\right)\left(-2\right)+3.75$
11

Multiply $\frac{1}{2}$ times $-1$

$\left({\left(-1\right)}^2\left(-0.5\right)+2^2\cdot 0.5\right)\left(-2\right)+3.75$
12

Calculate the power

$\left(1\left(-0.5\right)+4\cdot 0.5\right)\left(-2\right)+3.75$
13

Multiply $\frac{1}{2}$ times $4$

$\left(2-0.5\right)\left(-2\right)+3.75$
14

Subtract the values $2$ and $-\frac{1}{2}$

$1.5\left(-2\right)+3.75$
15

Multiply $-2$ times $\frac{3}{2}$

$3.75-3$
16

Subtract the values $\frac{15}{4}$ and $-3$

$\frac{3}{4}$

Answer

$\frac{3}{4}$

Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.21 seconds

Views:

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