Integrate �ç (x^3)/�ã((x^2)+4) dx
I think you want √(x^2+4) but I don't know what the �ç indicates.
copy/paste mangles various fonts
Here is what I meant to put:
Integrate �ç(x^3)/�ã((x^2)+4) dx
it keeps messing up...but you have the square root right and the ç was supposed to be the S shaped symbol.
so, you want
∫ x^3/√(x^2+4) dx
let u^2 = x^2+4
2u du = 2x dx
x^2 = u^2 - 4
∫ x^2/√(x^2+4) x dx
= ∫(u^2-4)/u u du
= ∫ u^2-4 du
= 1/3 u^3 - 4u
= 1/3 u (u^2 - 12)
= 1/3 √(x^2+4) (x^2 - 8)
To integrate the function ƒ(x) = (x^3) / √((x^2) + 4) dx, we can make a substitution to simplify the expression. Let's use the substitution u = x^2 + 4.
Differentiating both sides with respect to x, we get du/dx = 2x.
To solve for dx, we can rearrange this equation as dx = du / (2x).
Substituting this back into our original function, we have:
∫ (x^3) / √((x^2) + 4) dx = ∫ (x^3) / √u * (du / (2x)).
We can simplify this expression by canceling out the x terms:
∫ (x^3) / √((x^2) + 4) dx = ∫ (x^2) / 2√u du.
Now, let's simplify the integrand further by substituting x^2 for u - 4:
∫ (u - 4) / 2√u du.
Expanding this expression, we get:
∫ (u/2√u) - (4/2√u) du.
Simplifying the terms, we have:
∫ u^(1/2) / 2 - 2u^(-1/2) du.
Using the power rule of integration, we have:
= (2(u^(3/2)) / 3) - (4(2u^(1/2))) + C.
Substituting u back in terms of x, we have:
= (2(x^2 + 4)^(3/2) / 3) - (8(x^2 + 4)^(1/2)) + C.
Therefore, the integral of (x^3) / √((x^2) + 4) dx is equal to (2(x^2 + 4)^(3/2) / 3) - (8(x^2 + 4)^(1/2)) + C, where C is the constant of integration.