Integrate Using u-Substitution integral of (csc( square root of x)cot( square root of x))/( square root of x) with respect to x

Solution:

Step 1: Rewrite the integral using exponent rules

Step 1.1: Express $\sqrt{x}$ as $x^{1/2}$

$$\int \frac{\csc (x^{1/2})\cot (x^{1/2})}{x^{1/2}}dx$$

Step 1.2: Simplify the integrand by combining exponents

$$\int \csc (x^{1/2})\cot (x^{1/2})x^{-1/2}dx$$

Step 2: Perform u-substitution

Step 2.1: Let $u=x^{1/2}$, then find $du$

Step 2.1.1: Differentiate $u$ with respect to $x$

$$\frac{du}{dx}=\frac{1}{2}{x}^{-1/2}$$

Step 2.1.2: Solve for $dx$

$$dx=2x^{1/2}du$$

Step 2.2: Substitute $u$ and $dx$ into the integral

$$\int 2\csc (u)\cot (u)du$$

Step 3: Factor out constants from the integral

$$2\int \csc (u)\cot (u)du$$

Step 4: Integrate with respect to $u$

$$2(-\csc (u)+C)$$

Step 5: Simplify the expression

$$-2\csc (u)+C$$

Step 6: Back-substitute $u$ with $x^{1/2}$

$$-2\csc (x^{1/2})+C$$

Knowledge Notes:

The problem involves integrating a trigonometric function using u-substitution. The key knowledge points include:

1. Exponent Rules: Understanding how to rewrite expressions with radicals as fractional exponents is crucial. The rule $\sqrt[n]{a^x}=a^{x/n}$ is applied to convert $\sqrt{x}$ into $x^{1/2}$.

2. u-Substitution: This is a technique used in integration to simplify the integral by substituting a part of the integrand with a new variable $u$. The differential $du$ is then found by differentiating $u$ with respect to $x$.

3. Differentiation: Knowing how to differentiate power functions is important. The power rule states that $\frac{d}{dx}[x^n]=n{x}^{n-1}$.

4. Integration of Trigonometric Functions: Recognizing that the integral of $\csc (u)\cot (u)$ is $-\csc (u)$ is based on knowledge of trigonometric integrals.

5. Back-Substitution: After integrating with respect to $u$, it's necessary to substitute back in terms of the original variable $x$ to complete the problem.

6. Constants of Integration: When performing indefinite integration, a constant of integration $C$ is added to the result to account for the family of antiderivatives.

By applying these concepts, the integral can be evaluated step by step, leading to the final solution.