Integration Problem Homework Statement Solutions

[FallingMan]

Homework Statement

Problem 1:
[PLAIN]http://img266.imageshack.us/img266/3852/problem1m.jpg

Problem 2:
[PLAIN]http://img827.imageshack.us/img827/3862/problem3i.jpg


2. The attempt at a solution

Prob. 1:

Well for this one I rewrote the integral first.

$$\int sec(x)(1-sin^2(x))dx$$

Then I distributed the sec(x) term.

$$\int sec(x)-sec(x)sin^2(x)dx$$

I'm stuck here :(

Prob. 2:

I think I'm supposed to use u-sub here.

Let $$u = \sqrt{x}$$
$$du = \frac{1}{2\sqrt{x}}dx$$
Then solve for $$dx = 2\sqrt{x}du$$

Plug in for dx and you get:

$$\int (8x^2+2)(2\sqrt{x})(du)$$

I'm really not sure if this is what's supposed to happen...

Thanks,
fm

 

[eumyang]

Prob. 1:

Well for this one I rewrote the integral first.

∫sec(x) * (1-sin^2(x)) dx

Then I distributed the sec(x) term.

∫sec(x) - sec(x)*sin^2(x) dx

I'm stuck here :(

You're thinking too hard. Do you remember your trig identities? What does
$$1 - \sin^2 x$$ equal?

Prob. 2:

I think I'm supposed to use u-sub here.

Let u = sqrt(x), du = 1/2sqrt(x)dx
Then solve for dx = 2sqrt*du

Plug in for dx and you get:

∫(8x^2+2)*(2sqrt)*du

I'm really not sure if this is what's supposed to happen...

Thanks,
fm

Instead of doing that, write the fraction as a sum of two fractions:
$$\int \frac{8x^2 + 2}{\sqrt{x}} dx = \int \left( \frac{8x^2}{\sqrt{x}} + \frac{2}{\sqrt{x}} \right) dx = \int \frac{8x^2}{\sqrt{x}} dx + \int \frac{2}{\sqrt{x}} dx$$
... and integrate the fractions separately.

 

[FallingMan]

You're thinking too hard. Do you remember your trig identities? What does
$$1 - \sin^2 x$$ equal?

Okay, I just looked it up. 1-sin^2(x) is equivalent to cos^2(x), right? That makes it easier then..

$$\int \frac{cos^2(x)}{cos(x)}dx = \int cos(x)dx = sin(x) + C$$

:)?

Instead of doing that, write the fraction as a sum of two fractions:
$$\int \frac{8x^2 + 2}{\sqrt{x}} dx = \int \frac{8x^2}{\sqrt{x}}[ + \frac{2}{\sqrt{x}} dx$$
... and integrate the fractions separately.

Okay so first part would be...

$$\frac{16}{5}x^\frac{5}{2}$$

and second part would be...

$$4\sqrt{x}$$

So, adding them gives $$\frac{16}{5}x^\frac{5}{2}+4x^\frac{1}{2} + C$$

Thank for for the help (I hope I did them right )

By the way, how do you make the nice math symbols in your post? Any guide to how to do that?

EDIT: I think I got how to do it by viewing the edit options..

 

[eumyang]

#1 is right.

#2 is also right, but to nitpick, I personally don't like seeing an expression with fractional exponents and radicals mixed together. I would state the answer as
$$\frac{16}{5}x^{5/2} + 4x^{1/2} + C$$
(and don't forget the C! )

EDIT: It's good that you're trying LaTex, but you got some errors. Check the code I used. As for a guide, you could look "ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf"[/URL].

 

[FallingMan]

#1 is right.

#2 is also right, but to nitpick, I personally don't like seeing an expression with fractional exponents and radicals mixed together. I would state the answer as
$$\frac{16}{5}x^{5/2} + 4x^{1/2} + C$$
(and don't forget the C! )

EDIT: It's good that you're trying LaTex, but you got some errors. Check the code I used. As for a guide, you could look "ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf"[/URL].[/QUOTE]

Thank for your help - I'm a real math noob so I might make another thread b/c I may have some more questions, lol.

I might also edit this post to try some equations and things out. :)