Integrate this?

Question

integrate y^2* sqrt(1+y^3) dy from 0 to 1

Answer 1

use this site to calculate any integral problems you have :
http://integrals.wolfram.com/index.jsp

goodluck!

Answer 2

∫ (0 to 1, y^2 * sqrt(1 + y^3) dy )

To solve this, use substitution. To show the substitution in action, I'm first going to rearrange the expression.

∫ (0 to 1, sqrt(1 + y^3) y^2 dy )

Let u = 1 + y^3. Then
du = 3y^2 dy, so
(1/3)du = y^2 dy

Note that y^2 dy is the tail end of our integral, so (1/3)du will be the tail end of our new integral.

Also, our bounds for integration change.
When x = 0, u = 1 + 0^3 = 1 + 0 = 1
When x = 1, u = 1 + 1^3 = 1 + 1 = 2
So our bounds are 1 to 2.

∫ (1 to 2, sqrt(u) (1/3) du )

Factor the (1/3) from the integral.

(1/3) ∫ (1 to 2, sqrt(u) du )

Rewrite sqrt(u) as u^(1/2).

(1/3) ∫ (1 to 2, u^(1/2) du )

Use the reverse power rule to integrate. Remember that the integral of x^n (for n not equal to -1) is *1/[n + 1])x^(n + 1).

In our case, (1/2) + 1 = (3/2), which is offset by (2/3).

(1/3) [ (2/3)u^(3/2) ] {evaluated from 1 to 2 }

(1/3) [ (2/3)(2^(3/2)) ] - (1/3) [ (2/3)(1^(3/2) ]

This simplifies as

(2/9)*2^(3/2) - (2/9)*1

(2/9)*2^(3/2) - (2/9)

Answer 3

Use sustitution:

u = 1 + y^3
du = 3y^2dy

dy = du/(3y^2)

Replacing these values in the integral you'll have:

∫y^2sqrt(u)du / 3y^2

Cancelling the y's:

∫(1/3sqrt(u)du)

Integrating:

∫(1/3sqrt(u)du) = 1/3u^3/2 * 2/3 + C
= 2u^3/2 + C

Replacing back the original value:

= 2(1 + y^3)^3/2 + C

Now just evaluate the integral in its limits.

Answer 4

y^2* sqrt(1+y^3) dy

Let u=(1+y^3)
du=3y^2.dy

So now you need to integrate
∫(1/3)u^(1/2)du
=(1/3)/(3/2)u^(3/2)
=(2/9)u^(3/2)
=(2/9)(1+y^3)^3/2 with limits of 0 and 1

ie (2/9)(2^3/2 - 1)
=0.6285 (approx)

Answer 5

I = ∫ y² √(1 + y³) dy--------lims 0 to 1
Let u = 1 + y³
du/3 = y².dy
I = (1/3).∫ u^(1/2) du------lims 1 to 2
I = (2/9) u^(3/2)------lims 1 to 2
I = (2/9) [ 2^(3/2) - 1 ]
I = (2/9).[ 2.√2 - 1 ]

Answer 6

Use integration by parts !

Answer 7

it has limits, stick it in ur calculator.