# Thread: Integration with change of variable

1. ## Integration with change of variable

Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!

2. Originally Posted by aceband
Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!
Substitute $\displaystyle u = \sin x$ to get $\displaystyle \int 1 + u^2 \, du$ ....

3. Originally Posted by aceband
Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!
You're on the right track.

Note that if you let $\displaystyle u = \sin{x}$ then $\displaystyle \frac{du}{dx} = \cos{x}$.

So the integral becomes

$\displaystyle \int{\cos{x}(1 + \sin^2{x})\,dx} = \int{(1 + u^2)\,\frac{du}{dx}\,dx}$

$\displaystyle = \int{1 + u^2\,du}$

$\displaystyle = u + \frac{1}{3}u^3 + C$

$\displaystyle = \sin{x} + \frac{1}{3}\sin^3{x} + C$.
The text contains a mathematical question about integration and multiple responses providing a solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# Thread: Integration with change of variable

1. ## Integration with change of variable

Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!

2. Originally Posted by aceband
Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!
Substitute $\displaystyle u = \sin x$ to get $\displaystyle \int 1 + u^2 \, du$ ....

3. Originally Posted by aceband
Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?

$\displaystyle \int cos(x)(1+sin^2(x)) dx$

Thanks again!
You're on the right track.

Note that if you let $\displaystyle u = \sin{x}$ then $\displaystyle \frac{du}{dx} = \cos{x}$.

So the integral becomes

$\displaystyle \int{\cos{x}(1 + \sin^2{x})\,dx} = \int{(1 + u^2)\,\frac{du}{dx}\,dx}$

$\displaystyle = \int{1 + u^2\,du}$

$\displaystyle = u + \frac{1}{3}u^3 + C$

$\displaystyle = \sin{x} + \frac{1}{3}\sin^3{x} + C$.
user
The text contains a mathematical question about integration and multiple responses providing a solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER