How can I solve this standard integral using substitution?

[Andrea Vironda]

Homework Statement

Solve this integral

Relevant Equations

I think parametrization is needed

Hi,
I'd like to integrate this function: $$ \int _0^ {\pi/2} 2 \sin(x) \cos(x) \sqrt {1+\sin^{2}(x) } dx $$.
I think I should introduce some substitution but I'm not sure. How should I proceed?

 

[phyzguy]

I could tell you, but you will learn better if you try it on your own. What have you tried?

 

[Andrea Vironda]

In my opinion I could define $u=\sin^2(x)+1$, so $du=2\sin(x)\cos(x)dx$.
Then $\int_0^{\pi/2}\sqrt{u}du=2(\sqrt2-1)$ but this is not the right solution.

 

[docnet]

Don't forget to change the limits of integration, unless you substitute u back into the expression after integration.

This is what I'm getting for the corresponding indefinite integral.

If I may, I would suggest a step-by-step calculator to guide you.

https://www.integral-calculator.com

That calculator cannot help you on exams, but it could help you practice solving integrals.

 

[phyzguy]

That's a good substitution, but I don't think you did the integral correctly. What is the indefinite integral of u^(1/2)?

 

[etotheipi]

You don't need a substitution or anything, just note that $(1+\sin^2{x})' = 2\sin{x} \cos{x}$

 

[Andrea Vironda]

Solving integrals is an art