Problem 28 - Essential Calculus

Question

Verify the integration formula. $$ \int u^{n} \cos u d u=u^{n} \sin u-n \int u^{n-1} \sin u d u $$

Step-by-Step Solution

Answer

The formula has been verified correctly. The process involved the correct selection of 'u', 'dv', calculation of 'du', and 'v', applying the integration by parts formula and then combining all the parts to get the final formula.

Step 1: Identify u and dv

Choose \(u = u^n\) and \(dv = cos(u) du\). After this, calculate \(du = n u^{n-1} du\) and \(v = sin(u)\).

Step 2: Apply first part of the Integration by parts formula

We start with \(\int u dv = uv\), which we translate with our functions to \(\int u^n cos(u) du = u^n sin(u)\).

Step 3: Apply second part of the Integration by parts formula

Next, we need to compute \(- \int v du\) in the Integration by parts formula. Using our 'v' and 'du', this becomes \(- \int sin(u) * n u^{n-1} du = -n \int u^{n-1} sin(u) du\).

Step 4: Combine all parts

From Step 2 and Step 3, the entire formula becomes \(\int u^n cos(u) du = u^n sin(u) -n \int u^{n-1} sin(u) du\), which is what we aimed to verify.