Answer
$-\frac{18}{35}-\frac{2}{7}\left(\frac{3}{2}\right)^{7 / 2}+\frac{4}{5}\left(\frac{3}{2}\right)^{5 / 2} \approx0.509$
Work Step by Step
$$
\begin{aligned}
& \int_{5 \pi / 6}^{\pi} \frac{\cos ^{4} x}{\sqrt{1-\operatorname{sin} x}} d x \\
\\
& \text {First we will do the undefined integral}\\
\\
& \int \frac{\cos ^{4} x}{\sqrt{1-\operatorname{sin} x}} d x \\
& \int \frac{\cos ^{4} x \cdot \sqrt{1+\operatorname{sin} x}}{\sqrt{(1-\operatorname{sen} x)(1+\operatorname{sin} x)}} d x \\
& \int \frac{\cos ^{4} x \sqrt{1+\operatorname{sin} x}}{\sqrt{1-\operatorname{sin}^{2} x} d x} \\
& \cos ^{2} x+\operatorname{sin}^{2} x=1 \\
& \cos ^{2} x=1-\operatorname{sin}^{2} x \\
& \int \frac{\cos ^{4} x \sqrt{1+\operatorname{sin} x}}{\sqrt{\cos ^{2} x}} d x \rightarrow \int \frac{\cos ^{4} x \sqrt{1+\operatorname{sin} x}}{\cos x} d x \\
& \int \cos ^{3} x \sqrt{1+\operatorname{sin} x} d x \rightarrow \int \cos ^{2} x \cdot \cos x \cdot \sqrt{1+\operatorname{sin} x} d x \\
& \int\left(1-\operatorname{sin}^{2} x\right) \cos x \sqrt{1+\operatorname{sin} x} d x \\
& u=\operatorname{sin} x \quad d u=\cos x d x \quad d x=\frac{d u}{\cos x} \\
& \int\left(1-u^{2}\right) \sqrt{1+u} \cos x \frac{d u}{\cos x} \\
& \int\left(1-u^{2}\right) \sqrt{1+u} d u \\
& \int(u-1)(u+1) \sqrt{1+u} d u \\
& \int(u-1)(u+1)^{3 / 2} d u \\
& \begin{array}{l}
z=u+1 \quad d z=d u \quad u=z-1 \\
\int(z-2)(z)^{3 / 2} d z \rightarrow \int z^{5 / 2}-2 z^{3 / 2} d z
\end{array} \\
& \frac{2}{7} z^{7 / 2}-2\left(\frac{2}{5}\right) z^{5 / 2}+C \\
& \frac{2}{7}(u+1)^{7 / 2}-\frac{4}{5}(u+1)^{5 / 2}+C \\
& \frac{2}{7}(\operatorname{sin} x+1)^{7 / 2}-\frac{4}{5}(\operatorname{sin} x+1)^{5 / 2}+C
\end{aligned}
$$
$$
\begin{aligned}
& \int \frac{\cos ^{4} x}{\sqrt{1-\operatorname{sin} x}} d x=\frac{2}{7}(\operatorname{sin} x+1)^{7 / 2}-\frac{4}{5}(\operatorname{sin} x+1)^{5 / 2}+C \\
\\
& \text {Now we will evaluate the integral limits to find the answer}\\
\\
& \int_{\frac{5 \pi}{6}}^{\pi} \frac{\cos ^{4} x}{\sqrt{1-\operatorname{sin} x}} d x=\left[\frac{2}{7}(\operatorname{sin} x+1)^{7 / 2}-\frac{4}{5}(\operatorname{sin} x+1)^{5 / 2}\right]_{5 \pi / 6}^{\pi} \\
& =\left(\frac{2}{7}(\operatorname{sin} \pi+1)^{7 / 2}-\frac{4}{5}(\operatorname{sin} \pi+1)^{5 / 2}\right)-\left(\frac{2}{7}\left(\operatorname{sin} \frac{5 \pi}{6}+1\right)^{7 / 2}-\frac{4}{5}\left(\operatorname{sin} \frac{5 \pi}{6}+1\right)^{5 / 2}\right) \\
& =-\frac{18}{35}-\frac{2}{7}\left(\frac{3}{2}\right)^{7 / 2}+\frac{4}{5}\left(\frac{3}{2}\right)^{5 / 2} \\
& \approx0.509
\end{aligned}
$$
