integration of x^2/(x+3)sq.root of 3x+4 w.r.t. x
arrrgggghhh!
Wolfram says:
http://integrals.wolfram.com/index.jsp?expr=x%5E2%2F%28%28x%2B3%29%283x%2B4%29%5E%281%2F2%29%29&random=false
Hmmm. Consider
u^2 = 3x+4, so x = (u^2-4)/3
2u du = 3 dx
x+3 = (u^2+5)/3
x^2 = (u^2-4)^2/9
x^2 / (x+3)√(3x+4) dx
= 2(u^2-4) / 9(u^2+5) du
= (2u^2-26)/9 + 18/(u^2+5)
Integrate that to get
2/27 u (u^2-39) + 18/√5 arctan(u/√5)
= 2/27 √(3x+4)(3x-35) + 18/√5 arctan(u/√5)
w00t!
To integrate the given expression, let's break it down step by step.
Step 1: Rewrite the expression
Start by rearranging the expression to make it easier to integrate. Rewrite the denominator using the rules of exponents:
x^2/(x+3) * (3x+4)^0.5
Step 2: Simplify the expression
Next, simplify the expression by expanding and combining like terms:
x^2/(x+3) * sqrt(3x+4)
Step 3: Perform u-substitution
To integrate this expression, we can use the substitution method. Let's define a new variable, u, as the argument of the square root:
u = 3x + 4
Step 4: Find du/dx and solve for dx
Differentiate both sides of the equation with respect to x to find du/dx:
du/dx = 3
Solve for dx:
dx = du/3
Step 5: Rewrite the expression in terms of u
Now we can rewrite the expression in terms of u using the substitutions we found:
x^2/(x+3) * sqrt(3x+4) = (u - 4)/3 * sqrt(u)
Step 6: Integrate the expression with respect to u
Integrate the expression with respect to u using the power rule for integration:
∫ (u - 4)/3 * sqrt(u) du
= (1/3) * ∫ (u^(3/2) - 4u^(1/2)) du
= (1/3) * (2/5 * u^(5/2) - 8/3 * u^(3/2)) + C
Step 7: Substitute back in the original variable
Now we need to substitute back the original variable, x, in terms of u:
= (1/3) * (2/5 * (3x + 4)^(5/2) - 8/3 * (3x + 4)^(3/2)) + C
Therefore, the integral of x^2/(x+3)sqrt(3x+4) with respect to x is (1/3) * (2/5 * (3x + 4)^(5/2) - 8/3 * (3x + 4)^(3/2)) + C, where C is the constant of integration.