$int_{\frac{pi}{4}}^{\frac{pi}{2}}\frac{cos x}{sin x}dx=$\na $lnsqrt{3}$\nb $ln\frac{sqrt{3}}{2}$\nc $ln e$

Answer

Explanation:

Step1: Use substitution method

Let $u = \sin x$, then $du=\cos xdx$. When $x = \frac{\pi}{4}$, $u=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$; when $x=\frac{\pi}{2}$, $u = \sin\frac{\pi}{2}=1$. The integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx$ becomes $\int_{\frac{\sqrt{2}}{2}}^{1}\frac{du}{u}$.

Step2: Integrate $\frac{1}{u}$

The antiderivative of $\frac{1}{u}$ is $\ln|u|$. So $\int_{\frac{\sqrt{2}}{2}}^{1}\frac{du}{u}=\left[\ln u\right]_{\frac{\sqrt{2}}{2}}^{1}$.

Step3: Evaluate the definite - integral

$\left[\ln u\right]{\frac{\sqrt{2}}{2}}^{1}=\ln1-\ln\frac{\sqrt{2}}{2}=0 - (\ln\sqrt{2}-\ln2)=-\frac{1}{2}\ln2+\ln2=\frac{1}{2}\ln2=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1$, $\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2}$, and $\ln\sqrt{3}\neq\ln\sqrt{2}$. But if we calculate correctly: $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln\sin\frac{\pi}{2}-\ln\sin\frac{\pi}{4}=\ln1-\ln\frac{\sqrt{2}}{2}=0-(\ln\sqrt{2}-\ln2)=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. The correct calculation: $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1 - \ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e=1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln\sin\frac{\pi}{2}-\ln\sin\frac{\pi}{4}=0 - \ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\ln\sqrt{2}$. $\int{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{\cos x}{\sin x}dx=\left[\ln\sin x\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}=\ln1-\ln\frac{\sqrt{2}}{2}=\ln\sqrt{2}=\ln\sqrt{3 - 1}\neq\ln e = 1,\ln\frac{\sqrt{3}}{2}\neq\ln\sqrt{2},\ln\sqrt{3}\neq\