use the change of variables formula to evaluate the definite integral.\n\n$$ int _ { 0 } ^ { pi / 2 } cos ^…

use the change of variables formula to evaluate the definite integral.\n\n$$ int _ { 0 } ^ { pi / 2 } cos ^ { 4 } ( x ) sin ( x ) d x $$\n\n(use symbolic notation and fractions where needed.)\n\n$$ int _ { 0 } ^ { pi / 2 } cos ^ { 4 } ( x ) sin ( x ) d x = $$

use the change of variables formula to evaluate the definite integral.\n\n$$ int _ { 0 } ^ { pi / 2 } cos ^ { 4 } ( x ) sin ( x ) d x $$\n\n(use symbolic notation and fractions where needed.)\n\n$$ int _ { 0 } ^ { pi / 2 } cos ^ { 4 } ( x ) sin ( x ) d x = $$

Answer

Explanation:

Step1: Substitute (u = \cos(x))

Differentiate (u=\cos(x)), we get (du=-\sin(x)dx), so (\sin(x)dx=-du). When (x = 0), (u=\cos(0) = 1); when (x=\frac{\pi}{2}), (u=\cos(\frac{\pi}{2})=0). The integral (\int_{0}^{\frac{\pi}{2}}\cos^{4}(x)\sin(x)dx) becomes (-\int_{1}^{0}u^{4}du).

Step2: Evaluate the integral

Using the power - rule for integration (\int u^{n}du=\frac{u^{n + 1}}{n+1}+C(n\neq - 1)), we have (-\int_{1}^{0}u^{4}du=\int_{0}^{1}u^{4}du). [ \begin{align*} \int_{0}^{1}u^{4}du&=\left[\frac{u^{5}}{5}\right]_{0}^{1}\ &=\frac{1^{5}}{5}-\frac{0^{5}}{5} \end{align*} ]

Answer:

(\frac{1}{5})