1. ## Integrals

Hello!

I'm having some problems with Calculus. I can't even figure out how to get started, let alone do the problem :S

1)Find derivative of the function:

2)Calculate area enclosed by y^2 = 2x+6 and y = x -1

3) Calculate the integral
$
\int \frac {\sin2x}{1+cos^2x} dx
$

Thanks guys

2. Originally Posted by funnytim
Hello!

I'm having some problems with Calculus. I can't even figure out how to get started, let alone do the problem :S

1)Find derivative of the function:

2)Calculate area enclosed by y^2 = 2x+6 and y = x -1

3) Calculate the integral
$
\int \frac {\sin2x}{1+cos^2x} dx
$

Thanks guys

I'll give you a hand with the first one, you try to make some work of your own with the other ones:

Put $F(x):=\int\limits_{\tan x}^{x^2}\frac{1}{\sqrt{2+t^4}}dt$ As the function in the integral is continuous and defined everywhere, the Fundamental Theorem of integral Calculus tells us that

$F(x)=G(x^2)-G(\tan x)$ , where $G$ is a primitive function of the integrand function, but then:

$F'(x)=2xG'(x^2)-\frac{1}{\cos^2x}G'(\tan x)$ $=2x\,\frac{1}{\sqrt{2+x^8}}-\frac{1}{\cos^2x}\,\frac{1}{\sqrt{2+\tan^4x}}$

Tonio

3. For question three .

Consider $\sin(2x) = 2\sin(x) \cos(x)$

Then make a suitable substitution

4. 1.For your first question see attachment

2. For your second question see second attachment

3.For your third question write sin(2x) = 2sin(x)cos(x)

Then make the substitution u = cos(x) you will obtain -2u/(1+u^2)

in the integrand which you should be able to do

5. Thanks guys, you're great!

One question in#2 though. After i obtain:

How do I calculate it to obtain the answer, 18?

Thank you again!

6. Integrate and evaluate at the limits---submit your work if you don't get 18 and I'll see what happened

7. Actually, while I'm puzzling over that problem, here's another: How do I find a definite integral?

In particular:

Thanks again.

8. Originally Posted by funnytim
Actually, while I'm puzzling over that problem, here's another: How do I find a definite integral?

In particular:

Thanks again.

Doesn't it look suspiciously similar to a Riemann sum of the function $\frac{x}{x^2+1}$ over the interval $[0,1]$?

Tonio