Integrate x^2-2x from 0 to 2

\int_{0}^{2}\left(x^2-2x\right)dx

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Answer

$-\frac{4}{3}$

Step by step solution

Problem

$\int_{0}^{2}\left(x^2-2x\right)dx$
1

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

$\int_{0}^{2}-2xdx+\int_{0}^{2} x^2dx$
2

Taking the constant out of the integral

$\int_{0}^{2} x^2dx-2\int_{0}^{2} xdx$
3

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

$\left[\frac{x^{3}}{3}\right]_{0}^{2}-2\int_{0}^{2} xdx$
4

Evaluate the definite integral

$-2\int_{0}^{2} xdx+\frac{0^{3}}{3}\left(-1\right)+\frac{2^{3}}{3}$
5

Calculate the power

$-2\int_{0}^{2} xdx+\frac{0}{3}\left(-1\right)+\frac{8}{3}$
6

Divide $8$ by $3$

$-2\int_{0}^{2} xdx+0\left(-1\right)+\frac{8}{3}$
7

Any expression multiplied by $0$ is equal to $0$

$-2\int_{0}^{2} xdx+0+\frac{8}{3}$
8

Add the values $\frac{8}{3}$ and $0$

$\frac{8}{3}-2\int_{0}^{2} xdx$
9

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

$\frac{8}{3}-2\left[\frac{1}{2}x^2\right]_{0}^{2}$
10

Evaluate the definite integral

$\left(2^2\cdot 0.5-1\cdot 0^2\cdot 0.5\right)\left(-2\right)+2.6667$
11

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

$\left(0^2\left(-0.5\right)+2^2\cdot 0.5\right)\left(-2\right)+2.6667$
12

Calculate the power

$\left(0\left(-0.5\right)+4\cdot 0.5\right)\left(-2\right)+2.6667$
13

Any expression multiplied by $0$ is equal to $0$

$\left(0+4\cdot 0.5\right)\left(-2\right)+2.6667$
14

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

$\left(0+2\right)\left(-2\right)+2.6667$
15

Add the values $2$ and $0$

$2\left(-2\right)+2.6667$
16

Multiply $-2$ times $2$

$2.6667-4$
17

Subtract the values $\frac{8}{3}$ and $-4$

$-\frac{4}{3}$

Answer

$-\frac{4}{3}$

Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.22 seconds

Views:

72