∫ (4x+4)(x^2-2x-3)^2/3 dx
I suspect a typo. You probably meant
∫(4x-4)(x^2-2x-3)^(2/3) dx
or
∫(4x+4)(x^2+2x-3)^(2/3) dx
Assuming the first,
Let
u = x^2-2x-3
du = 2x-2 dx
Then you have
∫ 2u^(2/3) du
which is easy.
...
right?
If there is no typo, you are in deep trouble.
Hi Steve, thanks for your reply. Yes, it was a typo and your 2nd assumption was right. Please help, thanks!
∫(4x+4)(x^2+2x-3)^(2/3) dx
come on, man! It's still
∫2u^(2/3) du
= 2 (3/5) u^(5/3) + C
= 6/5 (x^2+2x-3)^(5/3) + C
To evaluate the given integral ∫ (4x+4)(x^2-2x-3)^2/3 dx, we can use a technique called u-substitution. Here's how you can proceed:
Step 1: Identify the substitution
Let's make the substitution u = x^2 - 2x - 3.
Step 2: Calculate the derivative of u
To calculate du (the differential of u), we take the derivative of u with respect to x:
du/dx = (d/dx) (x^2 - 2x - 3)
du/dx = 2x - 2
Step 3: Rearrange and solve for dx
Rearrange the equation from step 2 to solve for dx:
dx = (du) / (2x - 2)
Step 4: Substitute the variables
Substitute u = x^2 - 2x - 3 and dx = (du) / (2x - 2) into the original integral:
∫ (4x+4)(x^2-2x-3)^2/3 dx = ∫ (4(x^2 - 2x - 3) + 4)(x^2-2x-3)^2/3 dx
= ∫ (4u + 4) (u^(2/3)) (du) / (2x - 2)
Step 5: Simplify the expression
Combine like terms and simplify the expression further:
= ∫ (4u + 4) (u^(2/3)) (du) / (2(x - 1))
= ∫ (2u^(5/3) + 2u^(2/3)) (du) / (x - 1)
Step 6: Integrate with respect to u
Now, integrate the expression with respect to u:
= (2/3) ∫ u^(5/3) du + (2/3) ∫ u^(2/3) du
= (2/3) [(3/8)u^(8/3) + (3/5)u^(5/3)] + C
= (1/4)u^(8/3) + (1/5)u^(5/3) + C
Step 7: Substitute back the original variable
Replace u with the original expression x^2 - 2x - 3:
= (1/4)(x^2 - 2x - 3)^(8/3) + (1/5)(x^2 - 2x - 3)^(5/3) + C
Therefore, the final result of the integral is:
∫ (4x+4)(x^2-2x-3)^2/3 dx = (1/4)(x^2 - 2x - 3)^(8/3) + (1/5)(x^2 - 2x - 3)^(5/3) + C, where C is the constant of integration.