Problem 54 - Essential Calculus

Question

Integrate \(\int x \sqrt{4-x} d x\) (a) by parts, letting \(d v=\sqrt{4-x} d x\). (b) by substitution, letting \(u=4-x\).

Step-by-Step Solution

Answer

By parts, the integral evaluates to \(-\frac{8}{3} x^{\frac{3}{2}}+\frac{16}{5} x^{\frac{5}{2}}+C\), and by substitution, it evaluates to \(-\frac{16}{5}(4-x)^{\frac{5}{2}}+C\) .

Step 1 (a): Identify 'u' and 'dv'

In the integration by parts formula, choosing \(u = x\) and \(d v=\sqrt{4-x} d x\). After choosing the 'u' and 'dv', differentiate 'u' and integrate 'dv' to get 'du' and 'v' respectively.

Step 2 (a): Perform the Integration By Parts

With 'u', 'v', 'du' and 'dv' as determined, plug these into the integration by parts formula: \(\int{u dv}=u v - \int{v du}\). This will result in a new integral which is easier to evaluate.

Step 3 (a): Simplify and Evaluate

Simplify the result from step 2 until it becomes an arithmetic operation and can be evaluated surely.

Step 1 (b): Perform the substitution

In the substitution method, substituting \(u = 4 - x\) as given in the problem. Then calculate \(du\) and perform the substitution in the integral.

Step 2 (b): Simplify the Integral

After substitution, simplify the integral as possible.

Step 3 (b): Integrate

Integrate the simplified function after substitution.

Step 4 (b): Replace \(u\)

Finally, replace \(u\) with \(4-x\) and simplify, if possible.