Problem 21 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{x^{3}}{\sqrt{4-x^{2}}} d x $$

Step-by-Step Solution

Answer

The integral of \(\int \frac{x^{3}}{\sqrt{4-x^{2}}} d x = -1/2 (8\sqrt{4-x^2} - 2/3 (4-x^2)^{3/2} + C)\).

Step 1: Substitution

Start the solution by choosing \(u = 4 - x^2\). Then find the differential of \(u\), which is \(du = -2x dx\). Rewrite the integral in terms of \(u\), and for that divide both sides of \(du = -2x dx\) by \(-2x\), to obtain \( dx = -du/2x\). Substitute \(x^2\) from \(u = 4 - x^2\) to be \(x^2 = 4 - u\). Now replace \(dx\) and \(x\) in the integral to rewrite it in terms of \(u\). The integral becomes \(\int \frac{(4-u)(-du/2\sqrt{u}}{\sqrt{u}}\).

Step 2: Simplification

Simplify the integral after substitution. Degree of \(u\) in numerator and denominator will cancel out, so simplify it to \(-1/2 \int \frac{4-u}{\sqrt{u}} du\). This simplifies further to \(-1/2 \int (4u^{-1/2} - u^{1/2}) du\) by distributing the numerator across the denominator.

Step 3: Using the Power Rule for Integration

Now apply the power rule for integral, which states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), unless \(n = -1\). Use this rule to the each term in the integral \(-1/2 \int (4u^{-1/2} - u^{1/2}) du\), and we obtain \(-1/2 (8u^{1/2} - 2/3 u^{3/2} + C)\).

Step 4: Replace \(u\) with original variable

Replace \(u\) with the original variable \(x\). Therefore, the final answer is \(-1/2 (8\sqrt{4-x^2} - 2/3 (4-x^2)^{3/2} + C)\).