QuestionCalculate the integral $\int 2 x \cos \left(x^{2}-5\right) \, dx$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the integral of the function $x \cos(x^{}-5)$ . . We will use the method of substitution to solve this integral.

STEP 2

Choose a suitable substitution. In this case, we let $u = x^{2}-5$ .
$u = x^{2}-5$

STEP 3

Differentiate $u$ with respect to $x$ to find $du/dx$ .
$\frac{du}{dx} =2x$

STEP 4

Rearrange the above equation to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

STEP 5

Substitute $u$ and $dx$ into the original integral.
$\int2x \cos(u) \frac{du}{2x}$

STEP 6

implify the integral by cancelling out the $2x$ terms.
$\int \cos(u) du$

STEP 7

Now, we can integrate $\cos(u)$ with respect to $u$ .
$\int \cos(u) du = \sin(u) + C$

STEP 8

Finally, substitute $u = x^{2}-5$ back into the integral.
$\sin(x^{2}-5) + C$ So, the value of $\int2x \cos(x^{2}-5)$ is $\sin(x^{2}-5) + C$ .

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QuestionCalculate the integral $\int 2 x \cos \left(x^{2}-5\right) \, dx$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the integral of the function $x \cos(x^{}-5)$ . . We will use the method of substitution to solve this integral.

STEP 2

Choose a suitable substitution. In this case, we let $u = x^{2}-5$ .
$u = x^{2}-5$

STEP 3

Differentiate $u$ with respect to $x$ to find $du/dx$ .
$\frac{du}{dx} =2x$

STEP 4

Rearrange the above equation to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

STEP 5

Substitute $u$ and $dx$ into the original integral.
$\int2x \cos(u) \frac{du}{2x}$

STEP 6

implify the integral by cancelling out the $2x$ terms.
$\int \cos(u) du$

STEP 7

Now, we can integrate $\cos(u)$ with respect to $u$ .
$\int \cos(u) du = \sin(u) + C$

STEP 8

Finally, substitute $u = x^{2}-5$ back into the integral.
$\sin(x^{2}-5) + C$ So, the value of $\int2x \cos(x^{2}-5)$ is $\sin(x^{2}-5) + C$ .

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QuestionCalculate the integral ∫2xcos⁡(x2−5) dx\int 2 x \cos \left(x^{2}-5\right) \, dx∫2xcos(x2−5)dx.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the integral of the function xcos⁡(x−5)x \cos(x^{}-5)xcos(x−5). . We will use the method of substitution to solve this integral.

STEP 2

Choose a suitable substitution. In this case, we let u=x2−5u = x^{2}-5u=x2−5.u=x2−5u = x^{2}-5u=x2−5

STEP 3

Differentiate uuu with respect to xxx to find du/dxdu/dxdu/dx.dudx=2x\frac{du}{dx} =2xdxdu​=2x

STEP 4

Rearrange the above equation to express dxdxdx in terms of dududu.dx=du2xdx = \frac{du}{2x}dx=2xdu​

STEP 5

Substitute uuu and dxdxdx into the original integral.∫2xcos⁡(u)du2x\int2x \cos(u) \frac{du}{2x}∫2xcos(u)2xdu​

STEP 6

implify the integral by cancelling out the 2x2x2x terms.∫cos⁡(u)du\int \cos(u) du∫cos(u)du

STEP 7

Now, we can integrate cos⁡(u)\cos(u)cos(u) with respect to uuu.∫cos⁡(u)du=sin⁡(u)+C\int \cos(u) du = \sin(u) + C∫cos(u)du=sin(u)+C

STEP 8

Finally, substitute u=x2−5u = x^{2}-5u=x2−5 back into the integral.sin⁡(x2−5)+C\sin(x^{2}-5) + Csin(x2−5)+CSo, the value of ∫2xcos⁡(x2−5)\int2x \cos(x^{2}-5)∫2xcos(x2−5) is sin⁡(x2−5)+C\sin(x^{2}-5) + Csin(x2−5)+C.

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QuestionCalculate the integral $\int 2 x \cos \left(x^{2}-5\right) \, dx$ .

Assumptions1. We are asked to find the integral of the function $x \cos(x^{}-5)$ . . We will use the method of substitution to solve this integral.

Choose a suitable substitution. In this case, we let $u = x^{2}-5$ .
$u = x^{2}-5$

Differentiate $u$ with respect to $x$ to find $du/dx$ .
$\frac{du}{dx} =2x$

Rearrange the above equation to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

Substitute $u$ and $dx$ into the original integral.
$\int2x \cos(u) \frac{du}{2x}$

implify the integral by cancelling out the $2x$ terms.
$\int \cos(u) du$

Now, we can integrate $\cos(u)$ with respect to $u$ .
$\int \cos(u) du = \sin(u) + C$

Finally, substitute $u = x^{2}-5$ back into the integral.
$\sin(x^{2}-5) + C$ So, the value of $\int2x \cos(x^{2}-5)$ is $\sin(x^{2}-5) + C$ .