Integrate Using u-Substitution integral of (csc( square root of x)cot( square root of x))/( square root of x) with respect to x
The problem asks for the integration of a trigonometric function that involves the cosecant and cotangent of the square root of $x$over the square root of $x$. The method to be used for solving this integral is u-substitution, which is a technique used in calculus to simplify integration by making a substitution of variables. The problem essentially requires identifying an appropriate substitution (often denoted by $u$) that will make the integral easier to evaluate. Once the substitution is made, the integral is re-expressed in terms of the new variable $u$, integrated, and then the substitution is reversed to get back to the original variable $x$.
$\int \frac{csc \left(\right. \sqrt{x} \left.\right) cot \left(\right. \sqrt{x} \left.\right)}{\sqrt{x}} d x$
Answer
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:
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 $x^{1/2}$.
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 $x$.
Differentiation: Knowing how to differentiate power functions is important. The power rule states that $\frac{d}{dx}[x^n] = nx^{n-1}$.
Integration of Trigonometric Functions: Recognizing that the integral of $\csc(u)\cot(u)$ is $-\csc(u)$ is based on knowledge of trigonometric integrals.
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.
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.
