Integrate Using u-Substitution integral of sin(x)^2cos(x)^3 with respect to x
The question asks for the evaluation of a definite or indefinite integral of a trigonometric function sin(x)^2 * cos(x)^3 with respect to x, utilizing the method of u-substitution. U-substitution is a technique used in integral calculus to simplify an integral by substituting a part of the integrand with a new variable u. This technique often simplifies the integration process by transforming the integrand into a form that is easier to integrate. You are expected to identify an appropriate substitution, usually by letting u be a function inside the integrand that when differentiated, its derivative appears elsewhere in the integrand. Then, you would transform the integral in terms of u, carry out the integration, and finally substitute back to get the original variable x.
$\int \left(sin\right)^{2} \left(\right. x \left.\right) \left(cos\right)^{3} \left(\right. x \left.\right) d x$
Answer
Solution:
Step 1:
Extract $\cos^2(x)$ from the integral: $\int \sin^2(x) (\cos^2(x)\cos(x)) \, dx$
Step 2:
Apply the Pythagorean identity to $\cos^2(x)$: $\int \sin^2(x) ((1 - \sin^2(x))\cos(x)) \, dx$
Step 3:
Set $u = \sin(x)$. Then $du = \cos(x) \, dx$, which implies $dx = \frac{1}{\cos(x)} \, du$.
Step 3.1:
Define $u = \sin(x)$ and calculate $\frac{du}{dx}$.
Step 3.1.1:
Differentiate $\sin(x)$: $\frac{d}{dx}[\sin(x)]$
Step 3.1.2:
The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$: $\cos(x)$
Step 3.2:
Express the integral in terms of $u$ and $du$: $\int u^2 (1 - u^2) \, du$
Step 4:
Expand the expression $u^2 (1 - u^2)$: $\int (u^2 - u^4) \, du$
Step 5:
Simplify the integral.
Step 5.1:
Distribute $u^2$ across the terms: $\int (u^2 - u^4) \, du$
Step 5.2:
Apply the exponent rules.
Step 5.2.1:
Rewrite the expression: $\int (u^2 - u^4) \, du$
Step 5.2.2:
Combine like terms using the power rule for exponents: $\int (u^2 - u^4) \, du$
Step 5.2.3:
Simplify the expression: $\int (u^2 - u^4) \, du$
Step 5.3:
Rearrange the terms: $\int (u^2 - u^4) \, du$
Step 5.4:
Simplify the expression: $\int (u^2 - u^4) \, du$
Step 6:
Separate the integral into two parts: $\int u^2 \, du - \int u^4 \, du$
Step 7:
Integrate $u^2$ using the power rule: $\frac{1}{3}u^3 + C - \int u^4 \, du$
Step 8:
Factor out the constant from the integral: $\frac{1}{3}u^3 + C - \int u^4 \, du$
Step 9:
Integrate $u^4$ using the power rule: $\frac{1}{3}u^3 - \frac{1}{5}u^5 + C$
Step 10:
Combine the terms: $\frac{1}{3}u^3 - \frac{1}{5}u^5 + C$
Step 11:
Substitute $u$ back with $\sin(x)$: $\frac{1}{3}\sin^3(x) - \frac{1}{5}\sin^5(x) + C$
Knowledge Notes:
The process of solving the integral of $\sin^2(x)\cos^3(x)$ using $u$-substitution involves several key knowledge points:
$u$-Substitution: This technique is used to simplify integrals by changing the variable of integration to a new variable $u$. The choice of $u$ is typically a function inside the integral that, when differentiated, appears elsewhere in the integral.
Pythagorean Identity: The identity $\cos^2(x) + \sin^2(x) = 1$ is used to rewrite $\cos^2(x)$ as $1 - \sin^2(x)$. This is a common trigonometric identity that simplifies expressions involving sine and cosine functions.
Differentiation: The process of finding the derivative of a function. In this case, the derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$.
Integration: The process of finding the integral of a function. The power rule for integration states that $\int x^n \, dx = \frac{1}{n+1}x^{n+1} + C$ for any real number $n \neq -1$, where $C$ is the constant of integration.
Algebraic Manipulation: This includes expanding, factoring, and simplifying algebraic expressions, which is necessary to rewrite the integral in a form that is easier to integrate.
By applying these concepts, the original integral is transformed into a simpler form that can be integrated directly to find the solution.
