Question

In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 4 \cos ^{2} x d x, \quad \cos 2 x=1-2 \cos ^{2} x$$

Short answer

The integral of \(4 \cos ^{2} x dx\) is \(2x - \sin 2x + C\).

Step-by-step solution

Apply the trigonometric identity

First, expressing \(\cos ^{2} x\) in terms of \(\cos 2x\) using the given trigonometric identity, we get \(\cos ^{2} x = (1-\cos 2x)/2\). This simplifies the integral to \( \int 4 (1-\cos 2x)/2 dx= \int 2 (1-\cos 2x) dx\)

Break up the integral

Split this into two separate integrals. The integral of a sum is equal to the sum of the integrals. Thus, it can be written as: \( \int 2 dx - \int 2 \cos 2x dx\).

Reformulate using u-substitution

For the second integral, perform a \(u\)-substitution. Let \(u = 2x\). Therefore, \(du = 2 dx\). Applying this substitution, the integral become \(- \int \cos u du\).

Evaluate the Integrals

Evaluate each integral. \(\int 2 dx = 2x\), and \( -\int \cos u du = - \sin u\).

Substitute u back in terms of x

Replace u back into the equation with the original variable, \(2x\). This results in \(- \sin 2x\).

Combine the results

Combine these results to get the original result. The computed integral is: \(2x - \sin 2x + C\), where C is the constant of integration.