Question

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \cos ^{4} \frac{x}{2} d x $$

Short answer

\(\int \cos^4 \frac{x}{2} dx = \frac{3}{8}x + \frac{1}{2}\sin x + \frac{1}{4}\sin 2x + C\). When plotted, the antiderivative forms a wave pattern which is vertically translated based on the value of C.

Step-by-step solution

Set Up Integration

We first set up the integral \(\int \cos^4 \frac{x}{2} dx\). We have to integrate using Power Reduction formula. The power reduction formula for \(\cos^4\) is given by \(\cos^4 (x) = \frac{1}{8}(3 + 4\cos(2x) + \cos(4x))\). Substitute this into the integral.

Substitute

Now our integral looks like this: \(\int \frac{1}{8}(3 + 4\cos x + \cos 2x) dx\). We can take the constant out of the integral and integrate term by term. Use the fundamental integration formulas \( \int \cos u du = \sin u + C \) and \(\int du = u + C \)

Evaluate the Integral

Evaluating the integral results in \(\frac{3}{8}x + \frac{1}{2} \sin x + \frac{1}{4} \sin 2x + C\)

Plot the antiderivatives

Plot the antiderivative function for two different values of C. Let's take C = 0 and C = 1. Important to note when plotting, you're adjusting the vertical positioning of the graph with the constant C.