QuestionCalculate the integral: $\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx$

Studdy Solution

STEP 1

Assumptions1. We are asked to evaluate the definite integral of the function 4 \sinx from $0$ to $\frac{\pi}{4}$ . . We know the integral of $\sin x$ is $-\cos x$ .

STEP 2

First, we need to apply the integral to the function. The integral of $4 \sin2x$ can be solved by using the substitution method. Let's set $u =2x$ .
$u =2x$

STEP 3

Differentiate $u$ with respect to $x$ to find $du$ .
$du =2 dx$

STEP 4

olve the equation from3 for $dx$ .
$dx = \frac{du}{2}$

STEP 5

Substitute $u$ and $dx$ into the integral. The new integral becomes $\int4 \sin u \frac{du}{2}$

STEP 6

implify the integral.
$\int2 \sin u du$

STEP 7

Now, we can integrate $2 \sin u$ with respect to $u$ . The integral of $\sin u$ is $-\cos u$ , so the integral becomes $-2 \cos u$

STEP 8

Substitute $2x$ back in for $u$ .
$-2 \cos2x$

STEP 9

Now, we need to evaluate this from $$ to $\frac{\pi}{4}$. This is done by substituting these values in for $x$.
$-2 \cos2\left(\frac{\pi}{4}\right) - \left(-2 \cos2()\right)$

STEP 10

implify the expression.
$-2 \cos \left(\frac{\pi}{2}\right) - \left(-2 \cos0\right)$

STEP 11

Evaluate $\cos \left(\frac{\pi}{}\right)$ and $\cos0$ .
$-(0) - \left(-()\right)$

STEP 12

implify the expression to get the final answer.
$0 - (-2) =2$ The value of the integral $\int_{0}^{\frac{\pi}{4}}4 \sin2 x d x$ is $2$ .

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QuestionCalculate the integral: $\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx$

Studdy Solution

STEP 1

Assumptions1. We are asked to evaluate the definite integral of the function 4 \sinx from $0$ to $\frac{\pi}{4}$ . . We know the integral of $\sin x$ is $-\cos x$ .

STEP 2

First, we need to apply the integral to the function. The integral of $4 \sin2x$ can be solved by using the substitution method. Let's set $u =2x$ .
$u =2x$

STEP 3

Differentiate $u$ with respect to $x$ to find $du$ .
$du =2 dx$

STEP 4

olve the equation from3 for $dx$ .
$dx = \frac{du}{2}$

STEP 5

Substitute $u$ and $dx$ into the integral. The new integral becomes $\int4 \sin u \frac{du}{2}$

STEP 6

implify the integral.
$\int2 \sin u du$

STEP 7

Now, we can integrate $2 \sin u$ with respect to $u$ . The integral of $\sin u$ is $-\cos u$ , so the integral becomes $-2 \cos u$

STEP 8

Substitute $2x$ back in for $u$ .
$-2 \cos2x$

STEP 9

Now, we need to evaluate this from $$ to $\frac{\pi}{4}$. This is done by substituting these values in for $x$.
$-2 \cos2\left(\frac{\pi}{4}\right) - \left(-2 \cos2()\right)$

STEP 10

implify the expression.
$-2 \cos \left(\frac{\pi}{2}\right) - \left(-2 \cos0\right)$

STEP 11

Evaluate $\cos \left(\frac{\pi}{}\right)$ and $\cos0$ .
$-(0) - \left(-()\right)$

STEP 12

implify the expression to get the final answer.
$0 - (-2) =2$ The value of the integral $\int_{0}^{\frac{\pi}{4}}4 \sin2 x d x$ is $2$ .

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QuestionCalculate the integral: ∫0π44sin⁡2x dx\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx∫04π​​4sin2xdx

Studdy Solution

STEP 1

Assumptions1. We are asked to evaluate the definite integral of the function 4 \sinx from 000 to π4\frac{\pi}{4}4π​. . We know the integral of sin⁡x\sin xsinx is −cos⁡x-\cos x−cosx.

STEP 2

First, we need to apply the integral to the function. The integral of 4sin⁡2x4 \sin2x4sin2x can be solved by using the substitution method. Let's set u=2xu =2xu=2x.u=2xu =2xu=2x

STEP 3

Differentiate uuu with respect to xxx to find dududu.du=2dxdu =2 dxdu=2dx

STEP 4

olve the equation from3 for dxdxdx.dx=du2dx = \frac{du}{2}dx=2du​

STEP 5

Substitute uuu and dxdxdx into the integral. The new integral becomes∫4sin⁡udu2\int4 \sin u \frac{du}{2}∫4sinu2du​

STEP 6

implify the integral.∫2sin⁡udu\int2 \sin u du∫2sinudu

STEP 7

Now, we can integrate 2sin⁡u2 \sin u2sinu with respect to uuu. The integral of sin⁡u\sin usinu is −cos⁡u-\cos u−cosu, so the integral becomes−2cos⁡u-2 \cos u−2cosu

STEP 8

Substitute 2x2x2x back in for uuu.−2cos⁡2x-2 \cos2x−2cos2x

STEP 9

Now, we need to evaluate this from $$ to $\frac{\pi}{4}$. This is done by substituting these values in for $x$.−2cos⁡2(π4)−(−2cos⁡2())-2 \cos2\left(\frac{\pi}{4}\right) - \left(-2 \cos2()\right)−2cos2(4π​)−(−2cos2())

STEP 10

implify the expression.−2cos⁡(π2)−(−2cos⁡0)-2 \cos \left(\frac{\pi}{2}\right) - \left(-2 \cos0\right)−2cos(2π​)−(−2cos0)

STEP 11

Evaluate cos⁡(π)\cos \left(\frac{\pi}{}\right)cos(π​) and cos⁡0\cos0cos0.−(0)−(−())-(0) - \left(-()\right)−(0)−(−())

STEP 12

implify the expression to get the final answer.0−(−2)=20 - (-2) =20−(−2)=2The value of the integral ∫0π44sin⁡2xdx\int_{0}^{\frac{\pi}{4}}4 \sin2 x d x∫04π​​4sin2xdx is 222.

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QuestionCalculate the integral: $\int_{0}^{\frac{\pi}{4}} 4 \sin 2 x \, dx$

Assumptions1. We are asked to evaluate the definite integral of the function 4 \sinx from $0$ to $\frac{\pi}{4}$ . . We know the integral of $\sin x$ is $-\cos x$ .

First, we need to apply the integral to the function. The integral of $4 \sin2x$ can be solved by using the substitution method. Let's set $u =2x$ .
$u =2x$

Differentiate $u$ with respect to $x$ to find $du$ .
$du =2 dx$

olve the equation from3 for $dx$ .
$dx = \frac{du}{2}$

Substitute $u$ and $dx$ into the integral. The new integral becomes $\int4 \sin u \frac{du}{2}$

implify the integral.
$\int2 \sin u du$

Now, we can integrate $2 \sin u$ with respect to $u$ . The integral of $\sin u$ is $-\cos u$ , so the integral becomes $-2 \cos u$

Substitute $2x$ back in for $u$ .
$-2 \cos2x$

Now, we need to evaluate this from $$ to $\frac{\pi}{4}$. This is done by substituting these values in for $x$.
$-2 \cos2\left(\frac{\pi}{4}\right) - \left(-2 \cos2()\right)$

implify the expression.
$-2 \cos \left(\frac{\pi}{2}\right) - \left(-2 \cos0\right)$

Evaluate $\cos \left(\frac{\pi}{}\right)$ and $\cos0$ .
$-(0) - \left(-()\right)$

implify the expression to get the final answer.
$0 - (-2) =2$ The value of the integral $\int_{0}^{\frac{\pi}{4}}4 \sin2 x d x$ is $2$ .