Need help solving some integrals: int [(x + 2) x^3] dx, etc.

[PlanMeca]

Hello everyone. Google directed me here for my problems and as i looked around, I'm in the right place.

I have problems with some integrals. They came to the test but they are little bit more advanced than we studied in class. So i hope someone has little spare time and could help me solve them.

$\displaystyle \displaystyle 1.\, \int\,$$\displaystyle \dfrac{5\, dx}{x^4}\, +\,$\(\displaystyle \displaystyle \int\, x^2\, \dfrac{\sqrt[3]{\strut x\,}}{\sqrt{\strut x\,}}\, dx\)

$\displaystyle \displaystyle 2.\, \int\,$$\displaystyle (x\, +\, 2)\, x^3\, dx$

$\displaystyle \displaystyle 3.\,$$\displaystyle \dfrac{5x^2\, -\, 6x\, +\, 1}{x}\, dx$

$\displaystyle \displaystyle 4.\, \int_0^1\,$$\displaystyle \dfrac{x\, dx}{x^2\, +\, 1}$

$\displaystyle \displaystyle 5.\, \int_0^{\frac{\pi}{2}}\,$$\displaystyle e^{\cos(x)}\, \cdot\, \sin(x)\, dx$

Looking forward to your replies.
Thank you !

 

[ksdhart]

For the first problem, I'd begin by rewriting the two integrals being added. The left one, you can pull out a constant, and then you're left with 1/x⁴ which is easy enough to integrate. The right integral is a bit more difficult, but with simplification, it also becomes far easier. If you rewrite it as:

$\displaystyle \displaystyle \int \:x^2\cdot \frac{\sqrt[3]{x}}{\sqrt{x}}dx=\int \:x^2\cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{2}}}dx$

Now, if you recall your exponent rules, you should see how to proceed from here. For instance, x^a/x^b = ?? and x^a*x^b = ??

For the second problem, I'd distribute the x³ term across. Then you're left with two expressions of x being added. Do you recall a rule that can help you with that situation?

The third problem can be simplified by recalling your algebra. You can divide each term in the numerator individually by x. Then you're left with three expressions of x being added together. The same hint applies here as in problem 2.

For the fourth problem, I'd use u-substitution. What do you notice about the derivative of the denominator? How is that similar to the numerator? Does that give you a hint as to what you might use for the substitution?

The fifth problem can also be tackled with u-sub. Notice that e is being raised to the power of cosine x. What do you know about the derivatives of cosine and sine? What, then, does that suggest you use for the substitution?

 

[stapel]

I have problems with some integrals. They...are little bit more advanced than we studied in class....

$\displaystyle \displaystyle 1.\, \int\,$$\displaystyle \dfrac{5\, dx}{x^4}\, +\,$\(\displaystyle \displaystyle \int\, x^2\, \dfrac{\sqrt[3]{\strut x\,}}{\sqrt{\strut x\,}}\, dx\)

You applied what you learned back in algebra, converting 5/x^4 to 5x^(-4) and x^2 (cbrt(x)/sqrt(x)) to x^2 (x^(1/3)/x^(1/2) to x^2 x^(1/3-1/2) to x^(2 - 1/6), etc. Then you applied the Power Rule. Where did this lead? Where did you get stuck?

$\displaystyle \displaystyle 2.\, \int\,$$\displaystyle (x\, +\, 2)\, x^3\, dx$

You did the algebraic division, getting 1/x^2 + 2/x^3 = x^(-2) + 2x^(-3). You applied the Power Rule. Where did this lead? Where did you get stuck?

$\displaystyle \displaystyle 3.\,$$\displaystyle \dfrac{5x^2\, -\, 6x\, +\, 1}{x}\, dx$

Assuming that there was meant to be an "integral" symbol in front, what did you do, after dividing through by the x and apply the Power Rule?

$\displaystyle \displaystyle 4.\, \int_0^1\,$$\displaystyle \dfrac{x\, dx}{x^2\, +\, 1}$

You did the substitution, with u = x^2 + 1 so du = 2x dx and thus (1/2)du = x dx. Then what?

Please show all of your thoughts and efforts so far. Thank you!

 

[PlanMeca]

Thank you for your quick replies.

I managed to get this far, is it correct?

 

[ksdhart]

Okay, so, I'm not actually 100% sure what you did for problem #1. You started by pulling out the constant 5, leaving x^-4, but then it looks like you use z-substitution with z=-4. I'm not sure why you do this, as that leaves you with needing to take a partial integral at the end. That is to say, you'd need to integrate with respect to z and treat x as a constant, meaning:

$\displaystyle \int \:x^zdz=\frac{x^z}{log\left(x\right)}$

Instead of using z-substitution, I'd just use the standard rule. My book calls it the "anti-power rule," but yours might say something different. In any case, the rule is this:

$\displaystyle \int \:x^ndx=\frac{1}{n+1}x^{n+1}$

The rule works for any whole number, positive or negative (except -1), so:

$\displaystyle \int \:x^{-4}dx=\frac{1}{-4+1}x^{-4+1}=?$

The second integral from problem #1 looks good, except for a small arithmetic error. Specifically: $\displaystyle 2+\frac{1}{3}-\frac{1}{2}\ne \frac{5}{6}$. If it helps, try noting that 2 can be written as 12/6

Problems #2 and #3 look correct to me.

What you've done so far on problem #4 is fine as well. After that, you'd need to plug in your value for u, and then evaluate the integral on its bounds. But that shouldn't be too difficult, right?

And I don't see any work for problem #5, so I can't offer any more advice than my initial hint.