Integrate Using u-Substitution integral of sin(x)^2cos(x)^3 with respect to x

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:

1. $u$-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.

2. 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.

3. Differentiation: The process of finding the derivative of a function. In this case, the derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$.

4. 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 $n\ne -1$, where $C$ is the constant of integration.

5. 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.