Integrate sin(2x) from 0 to 2

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

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Answer

$\frac{\sqrt[3]{7}}{2}$

Step by step solution

Problem

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

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

$\left[-\frac{1}{2}\cos\left(2x\right)\right]_{0}^{2}$
2

Applying an identity of double-angle cosine

$\left[-\frac{1}{2}\left(1-2\sin\left(x\right)^2\right)\right]_{0}^{2}$
3

Applying a sine identity in order to reduce the exponent: $\displaystyle\sin(\theta)=\sqrt{\frac{1-\cos(2\theta)}{2}}$

$\left[-\frac{1}{2}\left(1-2\frac{1-\cos\left(2x\right)}{2}\right)\right]_{0}^{2}$
4

Evaluate the definite integral

$\left(\frac{\cos\left(2\cdot 2\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\left(-0.5\right)-1\cdot \left(\frac{\cos\left(0\cdot 2\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\left(-0.5\right)$
5

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

$\left(\frac{\cos\left(2\cdot 2\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\left(-0.5\right)-1\cdot \left(\frac{\cos\left(0\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\left(-0.5\right)$
6

Multiply $2$ times $2$

$\left(\frac{\cos\left(0\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\cdot 0.5+\left(\frac{\cos\left(4\right)\left(-1\right)+1}{2}\left(-2\right)+1\right)\left(-0.5\right)$
7

Calculating the cosine of $4$ degrees

$\left(\frac{1\left(-1\right)+1}{2}\left(-2\right)+1\right)\cdot 0.5+\left(\frac{1-0.6536\left(-1\right)}{2}\left(-2\right)+1\right)\left(-0.5\right)$
8

Multiply $-1$ times $-0.6536$

$\left(\frac{1-1}{2}\left(-2\right)+1\right)\cdot 0.5+\left(\frac{0.6536+1}{2}\left(-2\right)+1\right)\left(-0.5\right)$
9

Subtract the values $1$ and $-1$

$\left(\frac{0}{2}\left(-2\right)+1\right)\cdot 0.5+\left(\frac{0.6536+1}{2}\left(-2\right)+1\right)\left(-0.5\right)$
10

Add the values $1$ and $0.6536$

$\left(\frac{0}{2}\left(-2\right)+1\right)\cdot 0.5+\left(\frac{1.6536}{2}\left(-2\right)+1\right)\left(-0.5\right)$
11

Divide $\sqrt[3]{7}$ by $2$

$\left(0\left(-2\right)+1\right)\cdot 0.5+\left(0.8268\left(-2\right)+1\right)\left(-0.5\right)$
12

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

$\left(0+1\right)\cdot 0.5+\left(0.8268\left(-2\right)+1\right)\left(-0.5\right)$
13

Add the values $1$ and $0$

$1\cdot 0.5+\left(0.8268\left(-2\right)+1\right)\left(-0.5\right)$
14

Multiply $-2$ times $\frac{\sqrt[3]{7}}{2}$

$0.5+\left(1-1.6536\right)\left(-0.5\right)$
15

Subtract the values $1$ and $-\sqrt[3]{7}$

$0.5-0.6536\left(-0.5\right)$
16

Multiply $-\frac{1}{2}$ times $-0.6536$

$0.5+0.3268$
17

Add the values $\frac{67}{205}$ and $\frac{1}{2}$

$\frac{\sqrt[3]{7}}{2}$

Answer

$\frac{\sqrt[3]{7}}{2}$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.24 seconds

Views:

63