Evaluating definite integrals via substitution.

[shamieh]

Can someone make sure I'm on the right track with this problem? I'm a little confused because I thought that when you make a substitution you update the limits and get better numbers to work with when you plug them in the function in the end...Yet, it seems like I almost got worse numbers to work with.. Here is the problem, and what I have done so far.

Evaluate the following definite integral.

\(\displaystyle \int^1_0 36x^2(x^3 + 1)^4\)

\(\displaystyle u = x^3 + 1\)
\(\displaystyle du = 3x^2\)

But I don't have a \(\displaystyle 3x^2\) up top, I have a \(\displaystyle 36x^2\) so I divided out and got

\(\displaystyle \frac{du}{3} = x^2\)

now I update the limits and I get

\(\displaystyle 0^3 + 1 = 1\)
\(\displaystyle 1^3 + 1 = 2\)
so

\(\displaystyle \frac{1}{3} \int^2_1 du * u^4 = \frac{1}{3} * \frac{1}{5}u^5 = \frac{1}{15} * (2^3 + 1)^4] - [\frac{1}{15} * (1^3 + 1)^4]\)

Does this look correct? or have i messed up somewhere? I mean really i have to do \(\displaystyle 9^4\)? (sorry if that sounds ignorant, just want to make sure)

 

[alyafey22]

\(\displaystyle \frac{1}{3} \int^2_1 du * u^4 = \frac{1}{3} * \frac{1}{5}u^5 = \frac{1}{15} * (2^3 + 1)^4] - [\frac{1}{15} * (1^3 + 1)^4]\)

\(\displaystyle 36 \, \int^2_1 \frac{du}{3}\cdot \, u^4 =12 \int^2_1 \, du \cdot u^4\)

 

[MarkFL]

I would write the integral as:

\(\displaystyle 12\int_0^1\left(x^3+1 \right)^4\,3x^2\,dx\)

to get the result obtained by ZaidAlyafey.

 

[shamieh]

Seems I forgot to include the original 36 in the problem.

so I ended up with this:

\(\displaystyle [\frac{12}{5} (2^3 + 1)^5] - [\frac{12}{5} (1^3 + 1)^5]\)

\(\displaystyle [\frac{12}{5} * 59,049] - [\frac{12}{5} * 32]\)

\(\displaystyle [\frac{708,588}{5}] - [\frac{384}{5}] = \frac{708,204}{5}\)

This doesn't seem right though...Can anyone check my work?

 

[mathworker]

As you have not changed \(\displaystyle x\) to \(\displaystyle u\) you can't change limts from \(\displaystyle \int_0^1\) to \(\displaystyle \int_1^2\)

 

[shamieh]

As you have not changed \(\displaystyle x\) to \(\displaystyle u\) you can't change limts from \(\displaystyle \int_0^1\) to \(\displaystyle \int_1^2\)

?

- - - Updated - - -

Anyone else getting \(\displaystyle \frac{708204}{5}\) ?

 

[alyafey22]

\(\displaystyle 12 \int^2_1 \, du \cdot u^4 = \frac{12}{5} \left( 2^5-1\right) = \frac{31 \cdot 12}{5}=\frac{372}{5}\)

 

[shamieh]

\(\displaystyle 12 \int^2_1 \, du \cdot u^4 = \frac{12}{5} \left( 2^5-1\right) = \frac{31 \cdot 12}{5}=\frac{372}{5}\)

Zaid, is it because I changed the limits that I don't have to say, for example, [higher limit in the function] - [the lower limit in the function?]

In any other limit problem I would say [maximum limit plugged into antiderivative ] - [minimum plugged into antiderivative].. now you are just saying [maximum number plugged into anti derivative?]

 

[alyafey22]

Well, if you want to be in the safe side just find the anti-derivative. which is in your case

\(\displaystyle 12\frac{(x^3+1)^5}{5}+C\)

It remains just to use the FTC.

 

[shamieh]

Awesome, thanks.

Instead of

\(\displaystyle \frac{12}{5}(2)^5 - \frac{12}{5} (1)^5\)

I was doing \(\displaystyle \frac{12}{5}(2^3 + 1)^4\) - ...(Dull)