Problem 53 - Essential Calculus

Question

Integrate \(\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x\) (a) by parts, letting \(d v=\left(x / \sqrt{4+x^{2}}\right) d x\). (b) by substitution, letting \(u=4+x^{2}\).

Step-by-Step Solution

Answer

The integral can be evaluated using different techniques. Using integration by parts, the result is \(2x^2 \sqrt{4 + x^2} - 8/3 (4 + x^2)^{3/2} + C\). With substitution method, the result is \((1/3)(4 + x^2)^{3/2} + C\).

Step 1: Apply Integration by Parts for Part (a)

To apply integration by parts, you should first identify \(u\) and \(dv\). For this case, let \(u = x^2\) and \(dv = x / \sqrt{4 + x^2} dx\). Compute the derivatives and integrals: \(\ddot{u} = 2x dx\) and \(v = 2 \sqrt{4 + x^2}\). Then, proceed to use the formula of integration by parts given as \(\int u dv = uv - \int v du\). Substituting the identified components you get: \(x^2 * 2 \sqrt{4 + x^2} - \int 2 \sqrt{4 + x^2} * 2x dx\).

Step 2: Solve the Equation for Part (a)

The integral can be further simplified as: \(2x^2 \sqrt{4 + x^2} - 4 \int x \sqrt{4 + x^2} dx\). The remaining term is a standard form whose solution is known to be \(2/3 (4 + x^2)^{3/2}\). Thus, the solution is \(2x^2 \sqrt{4 + x^2} - 8/3 (4 + x^2)^{3/2} + C\).

Step 3: Apply Substitution for Part (b)

For the second part, a substitution can be done to simplify it. Let \(u = 4 + x^2\), then compute the derivatives: \(du =2x dx\). Rewriting the integral in terms of \(u\), and replacing the \(dx\) we have: \(\int x^3 / \sqrt{u} * d(u)/2x = \int u^{1/2} du / 2\).

Step 4: Solve the Equation for Part (b)

The integral can be directly computed. The integral of \(u^{1/2}\) with respect to \(u\) is \(2/3 u^{3/2}\). Substitute back \(u\) with \(4 + x^2\), thus the solution is \((1/3)(4 + x^2)^{3/2} + C\)