Integrate cos(x)^2 from 1 to 3

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

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Answer

$\frac{\sqrt{1}}{2}$

Step by step solution

Problem

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

Applying a cosine identity for reducing the exponent: $\displaystyle\cos(\theta)=\sqrt{\frac{1+\cos(2\theta)}{2}}$

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

Taking the constant out of the integral

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

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

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

The integral of a constant is equal to the constant times the integral's variable

$\frac{1}{2}\left(\int_{1}^{3}\cos\left(2x\right)dx+x\right)$
5

Apply the formula: $\int\cos\left(x\cdot a\right)dx$$=\frac{1}{a}\sin\left(x\cdot a\right)$, where $a=2$

$\left[\frac{1}{2}\left(\frac{1}{2}\sin\left(2x\right)+x\right)\right]_{1}^{3}$
6

Evaluate the definite integral

$\left(\sin\left(3\cdot 2\right)\cdot 0.5+3\right)\cdot 0.5-1\cdot \left(\sin\left(1\cdot 2\right)\cdot 0.5+1\right)\cdot 0.5$
7

Multiply $2$ times $3$

$\left(\sin\left(2\right)\cdot 0.5+1\right)\left(-0.5\right)+\left(\sin\left(6\right)\cdot 0.5+3\right)\cdot 0.5$
8

Calculating the sine of $6$ degrees

$\left(0.9093\cdot 0.5+1\right)\left(-0.5\right)+\left(3-0.2794\cdot 0.5\right)\cdot 0.5$
9

Multiply $\frac{1}{2}$ times $-\frac{19}{68}$

$\left(0.4546+1\right)\left(-0.5\right)+\left(3-0.1397\right)\cdot 0.5$
10

Subtract the values $3$ and $-\frac{19}{136}$

$\left(0.4546+1\right)\left(-0.5\right)+2.8603\cdot 0.5$
11

Add the values $1$ and $\frac{\sqrt[3]{3}}{3}$

$1.4546\left(-0.5\right)+2.8603\cdot 0.5$
12

Multiply $\frac{1}{2}$ times $\sqrt{8}$

$1.4301-0.7273$
13

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

$\frac{\sqrt{1}}{2}$

Answer

$\frac{\sqrt{1}}{2}$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.37 seconds

Views:

216