Given that the definite integral from $0$ to $\pi$ of ${\sin }^4\left(x\right)dx$ equals $\frac{3}{8\pi }$, what is the value of the definite integral from $0$ to $\pi$ of ${\cos }^4\left(x\right)dx$?

• A. $\frac{1}{2\pi }$
• B. $\frac{5}{8\pi }$
• C. $\frac{1}{4\pi }$
• D. $\frac{3}{8\pi }$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the Pythagorean identity: sin²(x) + cos²(x) = 1. This identity will help us relate the integrals of sin⁴(x) and cos⁴(x).
2. Step 2: Use the power-reduction formulas to express sin⁴(x) and cos⁴(x) in terms of sin²(x) and cos²(x). Specifically, sin⁴(x) = (sin²(x))² and cos⁴(x) = (cos²(x))².
3. Step 3: Note that the integral of sin⁴(x) + cos⁴(x) over [0, π] equals the integral of 1 over [0, π], because sin²(x) + cos²(x) = 1. This integral evaluates to π.
4. Step 4: Since the integral of sin⁴(x) over [0, π] is given as 3/8 π, use the relationship from Step 3 to find the integral of cos⁴(x). Specifically, the integral of cos⁴(x) = π - (integral of sin⁴(x)).
5. Step 5: Substitute the given value of the integral of sin⁴(x) (3/8 π) into the equation from Step 4 to compute the integral of cos⁴(x).