Math Help - Evaluating Integrals

1. Evaluating Integrals

Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

$\int_0^{\frac{3\pi}{2}}|sinx|dx$

Okay, so I know I need to split the integral because it is an absolute value. But I'm unsure of what values I need to use to do that. Can I just pick any value I want?

Also, I've been working on this problem, can someone tell me if I am approaching it correctly?

$g(x)=\int_{tanx}^{x^{2}}\frac{1}{\sqrt{2+t^4}}dt$

Here is what I have thus far:
$\int_{tanx}^{0}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{ x^{2}}\frac{1}{\sqrt{2+t^4}}dt$

$-\int_{0}^{tanx}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{ x^2}\frac{1}{\sqrt{2+t^4}}dt$

$g'(x)=\frac{-1}{\sqrt{2+tanx^4}}\frac{d}{dx}(tanx)+\frac{1}{\sq rt{2+x^6}}\frac{d}{dx}(x^2)$

..and that's all I have so far, but I'm not really sure what I need to do. Or if I'm even doing it correctly.

Thanks!!!!

2. Originally Posted by banshee.beat
Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

$\int_0^{\frac{3\pi}{2}}|sinx|dx$

Okay, so I know I need to split the integral because it is an absolute value. But I'm unsure of what values I need to use to do that. Can I just pick any value I want?

Mr F says: Draw the graph first by reflecting in the x-axis the parts of the graph of y = sin x that are below the x-axis. Then it should be quite obvious what the intervals need to be.

Also, I've been working on this problem, can someone tell me if I am approaching it correctly?

$g(x)=\int_{tanx}^{x^{2}}\frac{1}{\sqrt{2+t^4}}dt$

Here is what I have thus far:
$\int_{tanx}^{0}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{ x^{2}}\frac{1}{\sqrt{2+t^4}}dt$

$-\int_{0}^{tanx}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{ x^2}\frac{1}{\sqrt{2+t^4}}dt$

$g'(x)=\frac{-1}{\sqrt{2+tan{\color{red}^4} x}}\, \frac{d}{dx}(tanx)+\frac{1}{\sqrt{2+x^{\color{red} 8}}}\, \frac{d}{dx}(x^2)$

..and that's all I have so far, but I'm not really sure what I need to do. Or if I'm even doing it correctly. Mr F says: It's almost OK (see the mistakes I corrected in red). Obviously the next step is to calculate the derivatives ....

Thanks!!!!
..

3. Originally Posted by banshee.beat

Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

$\int_0^{\frac{3\pi}{2}}|sinx|dx$
$\int_{0}^{3\pi /2}{\left| \sin x \right|\,dx}=\int_{0}^{\pi }{\left| \sin x \right|\,dx}+\int_{\pi }^{3\pi /2}{\left| \sin x \right|\,dx},$ hence, the integral equals $\int_{0}^{\pi }{\sin x\,dx}-\int_{\pi }^{3\pi /2}{\sin x\,dx}.$