Integrate -1x^2-1x^2+2x from 1/4 to 3/4

\int_{\frac{1}{4}}^{\frac{3}{4}}\left(2x-x^2-x^2\right)dx

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Answer

$\frac{11}{48}$

Step by step solution

Problem

$\int_{\frac{1}{4}}^{\frac{3}{4}}\left(2x-x^2-x^2\right)dx$
1

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

$\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx+\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx+\int_{\frac{1}{4}}^{\frac{3}{4}}2xdx$
2

Adding $\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx$ and $\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx$

$\int_{\frac{1}{4}}^{\frac{3}{4}}2xdx+2\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx$
3

Taking the constant out of the integral

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+2\int_{\frac{1}{4}}^{\frac{3}{4}}-x^2dx$
4

Taking the constant out of the integral

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+2-\int_{\frac{1}{4}}^{\frac{3}{4}} x^2dx$
5

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

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+2\left[-\frac{x^{3}}{3}\right]_{\frac{1}{4}}^{\frac{3}{4}}$
6

Evaluate the definite integral

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+\left(\frac{\frac{3}{4}^{3}}{3}\left(-1\right)-1\cdot \left(\frac{\frac{1}{4}^{3}}{3}\right)\left(-1\right)\right)\cdot 2$
7

Multiply $-1$ times $-1$

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+\left(\frac{\frac{1}{4}^{3}}{3}\cdot 1+\frac{\frac{3}{4}^{3}}{3}\left(-1\right)\right)\cdot 2$
8

Calculate the power

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+\left(\frac{\frac{1}{64}}{3}\cdot 1+\frac{\frac{27}{64}}{3}\left(-1\right)\right)\cdot 2$
9

Divide $\frac{27}{64}$ by $3$

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+\left(\frac{1}{192}\cdot 1+\frac{9}{64}\left(-1\right)\right)\cdot 2$
10

Multiply $-1$ times $\frac{9}{64}$

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx+\left(\frac{1}{192}-\frac{9}{64}\right)\cdot 2$
11

Subtract the values $\frac{1}{192}$ and $-\frac{9}{64}$

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx-\frac{13}{96}\cdot 2$
12

Multiply $2$ times $-\frac{13}{96}$

$2\int_{\frac{1}{4}}^{\frac{3}{4}} xdx-\frac{13}{48}$
13

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

$2\left[\frac{1}{2}x^2\right]_{\frac{1}{4}}^{\frac{3}{4}}-\frac{13}{48}$
14

Evaluate the definite integral

$\left(0.75^2\cdot 0.5-1\cdot 0.25^2\cdot 0.5\right)\cdot 2-0.2708$
15

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

$\left(0.25^2\left(-0.5\right)+0.75^2\cdot 0.5\right)\cdot 2-0.2708$
16

Calculate the power

$\left(0.0625\left(-0.5\right)+0.5625\cdot 0.5\right)\cdot 2-0.2708$
17

Multiply $\frac{1}{2}$ times $\frac{9}{16}$

$\left(0.2812-0.0312\right)\cdot 2-0.2708$
18

Subtract the values $\frac{9}{32}$ and $-\frac{1}{32}$

$0.25\cdot 2-0.2708$
19

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

$0.5-0.2708$
20

Subtract the values $\frac{1}{2}$ and $-\frac{13}{48}$

$\frac{11}{48}$

Answer

$\frac{11}{48}$

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Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.23 seconds

Views:

114