Evaluate the Integral integral of cos(4x)^3 with respect to x

Problem

(∫_^)(cos(4*x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by splitting cos(4*x) into cos(4*x)*cos(4*x)

(∫_^)(cos(4*x)*cos(4*x)*d(x))

2. Apply the Pythagorean identity cos(θ)=1−sin(θ) to the squared term.

(∫_^)((1−sin(4*x))*cos(4*x)*d(x))

3. Perform a substitution by letting u=sin(4*x) which implies d(u)=4*cos(4*x)*d(x) or 1/4*d(u)=cos(4*x)*d(x)

(∫_^)((1−u2)1/4*d(u))

4. Distribute the constant and integrate the polynomial with respect to u

1/4*(∫_^)((1−u2)*d(u))

1/4*(u−(u3)/3)+C

5. Substitute back the original expression u=sin(4*x) into the result.

1/4*sin(4*x)−1/12*sin(4*x)+C

Final Answer

(∫_^)(cos(4*x)*d(x))=sin(4*x)/4−sin(4*x)/12+C

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