integral x^3/ (x^2 +1)^1/2
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x^3/(x^2+1)^(1/2)
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You must type function exactly in this form with brackets at (1/2)
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You will see solution step by step
let sqrt(x^2 + 1) = t
differentiating w.r.t x
then
x.dx / sqrt(x^2 + 1) = dt
substitution give
int(t^2 - 1)dt
= t^3/3 - t
= [(x^2 + 1)^3/2 ]/ 3 - (x^2 +1)^1/2
To find the integral of the function f(x) = x^3 / (x^2 + 1)^(1/2), we can use a technique called u-substitution.
Step 1: Let's start by making a substitution. Let u = x^2 + 1.
Step 2: Now, differentiate both sides of the equation to find du/dx. We know that du/dx = 2x.
Step 3: Rearrange the equation to solve for dx. dx = du / 2x.
Step 4: Now, we can rewrite the integral in terms of u and du:
∫ (x^3 / (x^2 + 1)^(1/2)) dx = ∫ (x^3 / u^(1/2)) (du / 2x)
Step 5: Simplify the expression:
= 1/2 ∫ (x^2 / u^(1/2)) du
Step 6: Notice that we can simplify x^2 / u^(1/2) to (u - 1) / u^(1/2).
= 1/2 ∫ ((u - 1) / u^(1/2)) du
Step 7: Integrate the expression with respect to u:
= 1/2 ∫ (u^(1/2) - u^(-1/2)) du
Step 8: Apply the power rule of integration to each term:
= 1/2 * (2/3 u^(3/2) - 2 u^(1/2)) + C
Step 9: Replace u with x^2 + 1:
= 1/3 (x^2 + 1)^(3/2) - (x^2 + 1)^(1/2) + C
Therefore, the integral of the function f(x) = x^3 / (x^2 + 1)^(1/2) is 1/3 (x^2 + 1)^(3/2) - (x^2 + 1)^(1/2) + C, where C is the constant of integration.