Integrate x^0.5-1x^2 from 0 to 1

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

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Answer

$\frac{1}{3}$

Step by step solution

Problem

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

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

$\int_{0}^{1}-x^2dx+\int_{0}^{1}\sqrt{x}dx$
2

Taking the constant out of the integral

$\int_{0}^{1}\sqrt{x}dx-\int_{0}^{1} x^2dx$
3

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

$\left[\frac{2}{3}\sqrt{x^{3}}\right]_{0}^{1}-\int_{0}^{1} x^2dx$
4

Evaluate the definite integral

$-\int_{0}^{1} x^2dx-1\cdot \sqrt{\left(0\right)^{3}}\cdot \frac{2}{3}+\sqrt{\left(1\right)^{3}}\cdot \frac{2}{3}$
5

Multiply $\frac{2}{3}$ times $-1$

$-\int_{0}^{1} x^2dx+\sqrt{\left(0\right)^{3}}\left(-\frac{2}{3}\right)+\sqrt{\left(1\right)^{3}}\cdot \frac{2}{3}$
6

Calculate the power

$-\int_{0}^{1} x^2dx+0\left(-\frac{2}{3}\right)+1\cdot \frac{2}{3}$
7

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

$-\int_{0}^{1} x^2dx+0+1\cdot \frac{2}{3}$
8

Multiply $\frac{2}{3}$ times $1$

$-\int_{0}^{1} x^2dx+0+\frac{2}{3}$
9

Add the values $\frac{2}{3}$ and $0$

$\frac{2}{3}-\int_{0}^{1} x^2dx$
10

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

$\left[-\frac{x^{3}}{3}\right]_{0}^{1}+\frac{2}{3}$
11

Evaluate the definite integral

$0.6667-1\cdot \left(\frac{0^{3}}{3}\right)\left(-1\right)+\frac{1^{3}}{3}\left(-1\right)$
12

Multiply $-1$ times $-1$

$0.6667+\frac{0^{3}}{3}\cdot 1+\frac{1^{3}}{3}\left(-1\right)$
13

Calculate the power

$0.6667+\frac{0}{3}\cdot 1+\frac{1}{3}\left(-1\right)$
14

Divide $1$ by $3$

$0.6667+0\cdot 1+0.3333\left(-1\right)$
15

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

$0.6667+0+0.3333\left(-1\right)$
16

Add the values $0$ and $\frac{2}{3}$

$0.3333\left(-1\right)+0.6667$
17

Multiply $-1$ times $\frac{1}{3}$

$0.6667-0.3333$
18

Subtract the values $\frac{2}{3}$ and $-\frac{1}{3}$

$\frac{1}{3}$

Answer

$\frac{1}{3}$

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Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.42 seconds

Views:

71