Problem 6 - Essential Calculus

Question

Use a table of integrals with forms involving the trigonometric functions to find the integral. $$ \int \frac{\cos ^{3} \sqrt{x}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral of \(\int \frac{\cos (\sqrt{x})^{3}}{\sqrt{x}} dx\) equals to \(\frac{2}{3}\sin(\sqrt{x}^{3}) + C\).

Step 1: Substitution

Use the substitution \(u = \sqrt{x}\), or equivalently \(x = u^2\), which implies that \(dx = 2udu\). Now replace \(x\) and \(dx\) in the original integral with the expressions involving \(u\).

Step 2: Simplify Integral

After substitution, the integral becomes \(\int 2u \cos(u^{3})du\). Notice that the integral is now in the form of \(cos(u^3)\times u\), which is equivalent to the derivative of \(\frac{1}{3} \sin(u^3)\). This observation allows the use of the standard integral form: \(\int cos(nu)du = \frac{1}{n} sin(nu) + C\), where \(n = u^2\).

Step 3: Final Calculation

Applying the formula, we obtain \(\int 2u \cos(u^{3})du = \frac{2}{3}\sin(u^{3}) + C\). Now replace the \(u\) by \(\sqrt{x}\) that was the initial substitution. The final answer is \(\frac{2}{3}\sin(\sqrt{x}^{3}) + C\).