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I understand the basics of anti-derivatives, and have been given the following questions:

1: $ \int [t^3 - 5e^t - 4\cos(t)] dt $

My answer:

$ \frac{t^4}{4} - e^{5t} + \sin(4t) + c $

2: $ \int [\frac{8}{x} + 5\sec(x)\tan(x)] dx$

My answer:

$8\ln(x) + \sec^2(5x) + c$

3: $ \int \frac{4x}{(x^2 +5)^3} dx $

My answer:

None yet, slightly confused since it's a quotient.

4: $ \int [x^2 \sqrt{2x^2 + 5}] dx $

My answer:

None yet, confused as well.

So I'm wondering if someone can check number 1 & 2, and then explain/show 3 & 4?

EDIT

These are what I get when I redo them (I've done my best to just look at suggestions rather than answers):

1: $ \frac{t^4}{4} - 5e^t - 4sin(t) + c $

2: $ 8ln(x) + 5sec(x) + c $

3: $ 2 \frac{(x^2 + 5)^{-2}}{-2}$

4: $ \frac{1}{6} \frac{(2x^3 + 5)^{3/2}}{3/2}$

  • 0
    To transform the integrand in 4 into a rational fraction of $t$ use the [Euler substitution](http://www.encyclopediaofmath.org/index.php/Euler_substitutions) $\sqrt{2x^2+5}=\sqrt{2}x+t.$2012-12-10

2 Answers 2

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$1$ and $2$ are not quite right. You seem to understand the idea but are missing one thing:

The derivative of for example $\sin(4t)$ by the chain rule would be $4 \cos(4t)$ not $4 \cos(t)$. This should show you why you shouldn't bring the constant inside the function you're taking the antiderivative of.

In general if you have a constant multiplied by a function you should take the antiderivative as if the constant were not there, and then multiply that by the constant.

For example for $4 \cos(t)$, we'll ignore the $4$ and take the antiderivative of $\cos(t)$ which is $\sin(t)$, then multiply it by $4$ to get $ 4 \sin(t).$ To check that this will work you should take the derivative and make sure its the what you want it to be!

You have made this error a couple if times in the first two and I advice you to try and find all the places you have and fix it.

For problem $3$ I advise you to try the $u$-sub $ u = x^2 + 5$.

And for problem $4$ I think the integral is intended to be $x^2\sqrt{2x^3 + 5}$ in which case I would suggest the $u$-sub $u = 2x^3 + 5$.

  • 0
    To transform the integrand in 4 as it is into a rational fraction of $t$ we can use the [Euler substitution](http://www.encyclopediaofmath.org/index.php/Euler_substitutions) $\sqrt{2x^2+5}=\sqrt{2}x+t.$2012-12-10
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First of all, you can always check whether an antiderivative is right by differentiating and checking whether you get what you started with.

For 1: Remember that $\int a f(x) \ dx = a \int f(x) \ dx$, that $\int e^x \ dx = e^x + C$ and that $(\sin x)' = \cos x$, not $-\cos x$. Also look at the derivative of $\sin(4x)$ and compare it with the derivative of $4 \sin x$.

For 2: What's the derivative of $\sec^2 x$, and what's the derivative of $\sec x$?

For 3: Try a substitution $u = x^2 + 5$.

For 4: Are you sure that's what you're supposed to do? That integral is pretty difficult, certainly not what I'd expect someone who's just starting with this to be able to do.

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    Thanks, I'll definitely be checking them with differentiation on my exam today! By the way, I added a few upvotes elsewhere on your answers as a thanks for the help.2012-12-10