Question

Derive the reduction formula \(\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x\).

Short answer

#tag_title#Reduction formula for integral of cosine function #tag_content#The reduction formula for the integral of the cosine function raised to the power of n is: \(\int \cos^n x dx = \frac{1}{n}\cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2}x dx\). This formula expresses the integral of cos^n(x) as a combination of a lower power of the cosine function and another integral with a reduced power. It is useful for simplifying and solving more complicated integrals involving the cosine function.

Step-by-step solution

Choose u and dv

In this step, we need to choose the functions u and dv to apply the integration by parts formula. Let \(u = \cos^{n-1}x\) and \(dv = \cos{x}dx\). This will allow us to simplify the integral more easily as we proceed.

Finding du and v

Now, we need to find the derivatives and integrals of u and dv, respectively. We have: \(du = (n-1)\cos^{n-2}x(-\sin x)dx = -(n-1)\cos^{n-2}x\sin x dx\) \(v = \int \cos{x}dx = \sin{x}\).

Apply integration by parts

Now, let's apply the integration by parts formula: \(\int \cos^n x dx = \int u dv = uv - \int v du\) Substituting the expressions we found in step 2: \(\int \cos^n x dx = \cos^{n-1}x \sin x + (n-1)\int \cos^{n-2}x \sin^2 x dx\).

Use the trigonometric identity

In this step, we will use the trigonometric identity \(\sin^2 x = 1 - \cos^2 x\) to simplify the integral: \(\int \cos^n x dx = \cos^{n-1}x \sin x + (n-1)\int \cos^{n-2}x (1 - \cos^2 x) dx\).

Separate the integrals

Now, let's distribute the integral over the sum inside the parentheses, and we get: \(\int \cos^n x dx = \cos^{n-1}x \sin x + (n-1)\int \cos^{n-2}x dx - (n-1)\int \cos^n x dx\).

Solve for the original integral

We can see that the integral we are looking for appears on both sides of the equation. To solve for it, we add \((n-1)\int \cos^n x dx\) to both sides. This gives: \(n \int \cos^n x dx = \cos^{n-1}x \sin x + (n-1)\int \cos^{n-2}x dx\). Finally, divide both sides by n: \(\int \cos^n x dx = \frac{1}{n}\cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2}x dx\). Now we have derived the reduction formula for the integral of the cosine function raised to the power of n.