How do you take the antiderivative of a function to a power? PLEASE!?

Question

For Example, take the anti derivative of (4x+1)^(1/2)

Answer 1

Integral (4x+1)^1/2 dx

4x + 1 = u

4 dx = du

dx = 1/4 du

Integral 1/4 (u)^1/2 du

1/4(Integral u^1/2 du)

1/4 (2/3u^3/2) + C

(1/6)(4x+1)^3/2 + C

Answer 2

You will need to do a U-substitution.

Let u = 4x + 1

Then du/dx = 4, so du = 4dx, thus dx = (1/4)du


The function then becomes u^(1/2)*(1/4)du

Take the antiderivative by using the power rule,

so it is (1/4)(1/2)u^(-1/2)+C = (1/8)(u^(-1/2)+C

Now replace u by (4x+1)

= (1/8)(4x+1)^(-1/2) + C = 1/(8sqrt(4x+1)) + C

Answer 3

Your exponent will be raised by one leaving you with ^(3/2). The first part will be (2x+(2/3)), so that when derived, from the power rule multiplied by (3/2), will equal (4x+1). It normally works to just divide your initial by what the exponent would be ^(n+1).

Answer 4

Hey there!

Here's the answer.

∫(4x+1)^(1/2) dx --> Write the problem.
∫sqrt(4x+1) dx --> Use the definition of square root.
u=4x+1, then du=4*dx --> Use substitution. Substitute u for 4x+1, such that du=4*dx or 1/4*du=dx.
∫1/4*sqrt(u) du --> Subsitute u for 4x+1 and 1/4*du for dx.
1/4∫sqrt(u) du --> Use the formula ∫a*f(x) dx=a∫f(x).
1/4∫u^(1/2) du --> Rewrite sqrt(u) as u^1/2.
1/4(u^(3/2)/(3/2))+C --> Use the formula ∫x^n dx=x^(n+1)/(n+1)+C, where n does not equal -1.
1/4*2/3*u^(3/2)+C --> Rewrite u^(3/2)/(3/2) as 2/3*u^(3/2).
1/6*u^(3/2)+C --> Multiply 1/4 and 2/3.
1/6*(4x+1)^(3/2)+C Substitute 4x+1 for u.

So the answer is 1/6*(4x+1)^(3/2)+C.

Hope it helps!

Answer 5

integral u^n du = u^(n+1)/(n+1) + C
In your case u = (4x+1)
du = 4 dx, so dx= du/4
so integral (4x+1)^1/2 dx = 1/4integral u^1/2du
= 1/4 u^(3/2)/(3/2)
= 1/6 u^3/2 + C = 1/6 (4x+1)^(3/2) + C

Answer 6

u=4x+1
du=4
4 integral u^1/2
= 4(4x+x)^(3/2) / 3/2

i could be wrong
i got a 3 in bc so don't count on me

Answer 7

use a scientific calculate dude

Answer 8

say what? idk what ur saying?