Anti-differentiation question?

Question

I'm stuck on this. Anyone who wants to help? :

∫(3x + 2)^4 dx

Answer 1

Let 3x+2 = u

3 dx = du

dx = du / 3

The given integral becomes

(1/3) ∫ u^4 du

=(1/3)u^5/5 +C

=u^5/15 +C

=(3x+2)^5 / 15 + C

Answer 2

Ok... so if you know the power rule, you'll have to do the same but backwards.  The best way is to try differentiating the equation first, just to see how it is you need to work backwards.

So pretend that the problem just said x^4.  You would integrate that as (1/5)x^5, right?  Because when you use the power rule, you multiply by the exponent, then subtract 1 from the exponenent, aka

(1/5)*5x^(5-1) = X^4

Now if you use the chain rule and the power rule, try differentiating the equation you have.  

(3x+2)^4 = 4(3x+2)^3   -- but here, since you are deriving 3x+2, you also have to multiply by the derivative of the inside (which is 3 for this case).  In the case of deriving x^4, you don't really see this because the derivative of x is just 1.

So now you see that deriving requires a multiplacation of the power, a multiplication of the derivative, and subtracting 1 from the power.

So now to integrate you just do the opposite of that:  devide by the power (which will be the power now +1), add 1 to the power, and divide by the derivative of the inside.

So take (3x+2)^4

Add 1 to the power

(3x+2)^5

divide by the new power - 5

(1/5)(3x+2)^5

divide by the derivative of the inside - 3

(1/15)(3x+2)^5

and there's your answer!

Hope it helps!

Answer 3

Multiply out the (3x + 2)^4, then integrate each of the terms separately.

Answer 4

Let u = 3x + 2

Then we get that:

du = 3dx

du/3 = dx

Therefore, your integral becomes:

(1/3) ∫ u⁴ du = u⁵/15 + C

But u = 3x + 2.

Therefore, we have that:

∫ (3x + 2)⁴ dx = (3x + 2)⁵/15 + C

Hope this helps!