- #1
tmwong
- 24
- 0
can anyone solve this equation?
∫x^3/(x^2+1)^(3/2) dx
∫x^3/(x^2+1)^(3/2) dx
NeutronStar said:[tex]\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx=
\frac{x^2+2}{\sqrt{x^2+1}}+C[/tex]
Integration is a mathematical process of finding the area under a curve. It involves finding the antiderivative of a function and calculating the area under the curve between two given points.
The formula for integration is ∫ f(x) dx = F(x) + C, where f(x) is the function being integrated and F(x) is its antiderivative. The constant C is known as the constant of integration.
The given integration problem is asking for the antiderivative of the function f(x) = x^3/(x^2+1)^(3/2) with respect to x.
To solve this integration problem, we can use the substitution method. Let u = x^2+1, then du/dx = 2x and dx = du/2x. Substituting into the original problem, we get ∫x^3/(x^2+1)^(3/2) dx = ∫x^2/(x^2+1)^(3/2) (du/2x) = ∫u^(-3/2) (du/2) = u^(-1/2) + C = (x^2+1)^(-1/2) + C.
Yes, there are other methods to solve this integration problem such as using trigonometric substitutions or partial fractions. However, the substitution method is the most straightforward and efficient method for this particular problem.