Finding antiderivative of (2x² + 1)⁷

[Nope]

Homework Statement

$$\int(2x^2+1)^7$$

Homework Equations

The Attempt at a Solution

u=2x^2+1
du=4xdx
u⁷ (1/4x)du
I am stuck... I don't know what to do next...

 

[Mark44]

Homework Statement

$$\int(2x^2+1)^7$$

Try to remember to put in the differential...

Homework Equations

The Attempt at a Solution

u=2x^2+1
du=4xdx
u⁷ (1/4x)du
I am stuck... I don't know what to do next...

Substitution won't work in this case, which you already found out. If you expand $\int(2x^2+1)^7$, you'll get a polynomial that you can integrate pretty easily.

 

[Nope]

wow, so i have to expand everything out? (2x^2+1)^7
that's a lot
is there any other way to do it?

 

[vela]

The x² suggests a trig substitution might work.

Edit: Actually, that looks to be more of a pain than just multiplying the polynomial out.

Hint: Use the binomial theorem.

 

[Mark44]

Nope, I don't think so, at least no way that's not a lot more complicated.

 

[elect_eng]

wow, so i have to expand everything out? (2x^2+1)^7
that's a lot
is there any other way to do it?

Is using a symbolic processor cheating? From Maxima ...

ratsimp((2*x^2+1)^7);

$$128\,{x}^{14}+448\,{x}^{12}+672\,{x}^{10}+560\,{x}^{8}+280\,{x}^{6}+84\,{x}^{4}+14\,{x}^{2}+1$$

 

[Mark44]

Probably not, but will you have one available during a test?

 

[Nope]

No , I don't think so.

 

[HallsofIvy]

It does help to know the "binomial theorem":

$$(a+ b)^n= \sum_{i=0}^n \begin{pmatrix}n \\ i\end{pmatrix}a^{i}b^{n-i}$$