Math / Calculus

QuestionCompute the integral $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=$ $\square$ $\text { 마온 }+C$ where $C$ is the constant of integration. Do not include the constant of integration in yo

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a function that involves e to the x and sine squared of e to the x. Watch out! It's easy to get tangled up with the e to the x inside the sine function, so let's be super careful with our substitution!

STEP 2

1. Substitute
2. Integrate
3. Back-substitute

STEP 3

Let's define our substitution!
We'll let $u = e^x$ .
This is because $e^x$ appears in both the numerator and within the sine function in the denominator.

STEP 4

Now, we need to find $\frac{du}{dx}$ .
The derivative of $e^x$ with respect to $x$ is simply $e^x$ .
So, $\frac{du}{dx} = e^x$ .

STEP 5

We can rewrite this as $du = e^x dx$ .
This will be perfect for substituting into our integral!

STEP 6

Let's rewrite our integral using the substitution: $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} dx = \int \frac{1}{\sin^2(u)} du$ Look how much cleaner that looks!

STEP 7

We know that $\frac{1}{\sin^2(u)}$ is the same as $\csc^2(u)$ .
So, our integral becomes $\int \csc^2(u) du$ .

STEP 8

Remember that the integral of $\csc^2(u)$ is $-\cot(u)$ .
So, we have: $\int \csc^2(u) du = -\cot(u)$

STEP 9

Now, let's substitute back in our original value for $u$ , which was $e^x$ .
This gives us our final result for the integral: $-\cot(e^x)$

STEP 10

Our final answer is $-\cot(e^x) + C$ , where C is the constant of integration.
Remember, we always add the constant of integration when we're dealing with indefinite integrals!

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QuestionCompute the integral $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=$ $\square$ $\text { 마온 }+C$ where $C$ is the constant of integration. Do not include the constant of integration in yo

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a function that involves e to the x and sine squared of e to the x. Watch out! It's easy to get tangled up with the e to the x inside the sine function, so let's be super careful with our substitution!

STEP 2

1. Substitute
2. Integrate
3. Back-substitute

STEP 3

Let's define our substitution!
We'll let $u = e^x$ .
This is because $e^x$ appears in both the numerator and within the sine function in the denominator.

STEP 4

Now, we need to find $\frac{du}{dx}$ .
The derivative of $e^x$ with respect to $x$ is simply $e^x$ .
So, $\frac{du}{dx} = e^x$ .

STEP 5

We can rewrite this as $du = e^x dx$ .
This will be perfect for substituting into our integral!

STEP 6

Let's rewrite our integral using the substitution: $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} dx = \int \frac{1}{\sin^2(u)} du$ Look how much cleaner that looks!

STEP 7

We know that $\frac{1}{\sin^2(u)}$ is the same as $\csc^2(u)$ .
So, our integral becomes $\int \csc^2(u) du$ .

STEP 8

Remember that the integral of $\csc^2(u)$ is $-\cot(u)$ .
So, we have: $\int \csc^2(u) du = -\cot(u)$

STEP 9

Now, let's substitute back in our original value for $u$ , which was $e^x$ .
This gives us our final result for the integral: $-\cot(e^x)$

STEP 10

Our final answer is $-\cot(e^x) + C$ , where C is the constant of integration.
Remember, we always add the constant of integration when we're dealing with indefinite integrals!

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QuestionCompute the integral ∫exsin⁡2(ex)dx=\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=∫sin2(ex)ex​dx= □\square□ 마온 +C\text { 마온 }+C 마온 +C where CCC is the constant of integration. Do not include the constant of integration in yo

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a function that involves *e* to the *x* and sine squared of *e* to the *x*. Watch out! It's easy to get tangled up with the *e* to the *x* inside the sine function, so let's be super careful with our substitution!

STEP 2

1. Substitute2. Integrate3. Back-substitute

STEP 3

Let's **define** our substitution!We'll let u=exu = e^xu=ex.This is because exe^xex appears in both the numerator and within the sine function in the denominator.

STEP 4

Now, we need to find dudx\frac{du}{dx}dxdu​.The derivative of exe^xex with respect to xxx is simply exe^xex.So, dudx=ex\frac{du}{dx} = e^xdxdu​=ex.

STEP 5

We can rewrite this as du=exdxdu = e^x dxdu=exdx.This will be perfect for substituting into our integral!

STEP 6

Let's **rewrite** our integral using the substitution: ∫exsin⁡2(ex)dx=∫1sin⁡2(u)du\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} dx = \int \frac{1}{\sin^2(u)} du∫sin2(ex)ex​dx=∫sin2(u)1​du Look how much cleaner that looks!

STEP 7

We know that 1sin⁡2(u)\frac{1}{\sin^2(u)}sin2(u)1​ is the same as csc⁡2(u)\csc^2(u)csc2(u).So, our integral becomes ∫csc⁡2(u)du\int \csc^2(u) du∫csc2(u)du.

STEP 8

Remember that the integral of csc⁡2(u)\csc^2(u)csc2(u) is −cot⁡(u)-\cot(u)−cot(u).So, we have: ∫csc⁡2(u)du=−cot⁡(u)\int \csc^2(u) du = -\cot(u)∫csc2(u)du=−cot(u)

STEP 9

Now, let's **substitute** back in our original value for uuu, which was exe^xex.This gives us our **final result** for the integral: −cot⁡(ex)-\cot(e^x)−cot(ex)

STEP 10

Our final answer is −cot⁡(ex)+C-\cot(e^x) + C−cot(ex)+C, where *C* is the constant of integration.Remember, we always add the constant of integration when we're dealing with indefinite integrals!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionCompute the integral $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} d x=$ $\square$ $\text { 마온 }+C$ where $C$ is the constant of integration. Do not include the constant of integration in yo

What is this asking? We need to find the integral of a function that involves e to the x and sine squared of e to the x. Watch out! It's easy to get tangled up with the e to the x inside the sine function, so let's be super careful with our substitution!

1. Substitute
2. Integrate
3. Back-substitute

Let's define our substitution!
We'll let $u = e^x$ .
This is because $e^x$ appears in both the numerator and within the sine function in the denominator.

Now, we need to find $\frac{du}{dx}$ .
The derivative of $e^x$ with respect to $x$ is simply $e^x$ .
So, $\frac{du}{dx} = e^x$ .

We can rewrite this as $du = e^x dx$ .
This will be perfect for substituting into our integral!

Let's rewrite our integral using the substitution: $\int \frac{e^{x}}{\sin ^{2}\left(e^{x}\right)} dx = \int \frac{1}{\sin^2(u)} du$ Look how much cleaner that looks!

We know that $\frac{1}{\sin^2(u)}$ is the same as $\csc^2(u)$ .
So, our integral becomes $\int \csc^2(u) du$ .

Remember that the integral of $\csc^2(u)$ is $-\cot(u)$ .
So, we have: $\int \csc^2(u) du = -\cot(u)$

Now, let's substitute back in our original value for $u$ , which was $e^x$ .
This gives us our final result for the integral: $-\cot(e^x)$

Our final answer is $-\cot(e^x) + C$ , where C is the constant of integration.
Remember, we always add the constant of integration when we're dealing with indefinite integrals!