How Do You Evaluate and Differentiate Complex Trigonometric Functions?

[Ivan1]

Evaluate ∫[sin2x/(1+(cos)^2 x) dx]Differentiate f(x) = (sin)^2 (e^((sin^2) x))

Hello, I'm just really stumped with these review questions and i have a test coming up. For the first, I'm not too sure what to do since there is a sin2x in general and for the second i don't know how to deal the the exponential. Some help would be really appreciated!

 

[Dethrone]

Hello Ivan, welcome to MHB! (Wave)

When you are integrating, it is always helpful to first see if you can find a function and its derivative in the integrand. In this case, rewriting the integrand produces exactly what we want:
$$\int \frac{\sin\left({2x}\right)}{1+\cos^2\left({x}\right)} \,dx=\int \frac{2\sin\left({x}\right)\cos\left({x}\right)}{1+1-\sin^2\left({x}\right)} \,dx$$

For your derivative, do you mean this?
$$\sin^2\left({e^{\sin^2x}x}\right)$$

Recall that given a function $e^{f(x)}$, its derivative is $e^{f(x)}\cdot f'(x)$

 

[topsquark]

Evaluate ∫[sin2x/(1+(cos)^2 x) dx]Differentiate f(x) = (sin)^2 (e^((sin^2) x))

Just to be clear, are your questions
\(\displaystyle \int \frac{\sin(2x)}{1 + \cos^2(x)}~dx\)

and
\(\displaystyle f(x) = \sin^2 \left ( e^{\sin^2(x)} \right )\)

-Dan

 

[Ivan1]

Just to be clear, are your questions
\(\displaystyle \int \frac{\sin(2x)}{1 + \cos^2(x)}~dx\)

and
\(\displaystyle f(x) = \sin^2 \left ( e^{\sin^2(x)} \right )\)

-Dan

Yes

 

[MarkFL]

Let's look at the first problem, and use a method similar to that suggested by Rido12:

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

Now, if we let:

\(\displaystyle u=1+\cos^2(x)\)

then we find:

\(\displaystyle du=2\cos(x)(-\sin(x))\,dx=-\sin(2x)\,dx\)

Now, if we write the integral as:

\(\displaystyle -\int\frac{-\sin(2x)\,dx}{1+\cos^2(x)}\)

What does it become when we use our $u$-substitution?

 

[topsquark]

\(\displaystyle f(x) = \sin^2 \left ( e^{\sin^2(x)} \right )\)

Do this with the chain rule. I'll separate the different derivatives with [ ] symbols.
\(\displaystyle f'(x) = \left [ 2~sin \left ( e^{\sin^2(x)} \right )~cos \left ( e^{\sin^2(x)} \right ) \right ] \cdot \left [ e^{\sin^2(x)} \right ] \cdot \left [ 2~sin(x)~cos(x) \right ] \)

-Dan