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

Solution:

Step 1:

Transform ${\sin }^2(x)$ using the half-angle identity to $\frac{1-\cos (2x)}{2}$. Integrate $\int \frac{1-\cos (2x)}{2}dx$.

Step 2:

Extract the constant $\frac{1}{2}$ from the integral to simplify. $\frac{1}{2}\int (1-\cos (2x))dx$.

Step 3:

Decompose the integral into two separate integrals. $\frac{1}{2}(\int dx-\int \cos (2x)dx)$.

Step 4:

Integrate the constant function. $\frac{1}{2}(x+C-\int \cos (2x)dx)$.

Step 5:

Factor out the negative sign in front of the cosine integral. $\frac{1}{2}(x+C-\int \cos (2x)dx)$.

Step 6:

Perform a u-substitution where $u=2x$, thus $du=2dx$ and $dx=\frac{1}{2}du$.

Step 6.1:

Define the substitution $u=2x$ and calculate $\frac{du}{dx}$.

Step 6.1.1:

Differentiate $2x$ to find $du$. $\frac{d}{dx}(2x)$.

Step 6.1.2:

Apply the constant multiple rule in differentiation. $2\frac{d}{dx}(x)$.

Step 6.1.3:

Use the Power Rule, where the derivative of $x^n$ is $n{x}^{n-1}$ with $n=1$. $2\cdot 1$.

Step 6.1.4:

Simplify the derivative. $2$.

Step 6.2:

Rewrite the integral in terms of $u$ and $du$. $\frac{1}{2}(x+C-\int \cos (u)\frac{1}{2}du)$.

Step 7:

Combine the cosine function with the constant $\frac{1}{2}$. $\frac{1}{2}(x+C-\int \frac{\cos (u)}{2}du)$.

Step 8:

Extract the constant $\frac{1}{2}$ from the integral. $\frac{1}{2}(x+C-\frac{1}{2}\int \cos (u)du)$.

Step 9:

Integrate $\cos (u)$ with respect to $u$. $\frac{1}{2}(x+C-\frac{1}{2}(\sin (u)+C))$.

Step 10:

Simplify the expression. $\frac{1}{2}(x-\frac{1}{2}\sin (u))+C$.

Step 11:

Substitute back $u=2x$. $\frac{1}{2}(x-\frac{1}{2}\sin (2x))+C$.

Step 12:

Final simplification.

Step 12.1:

Combine the sine function with $\frac{1}{2}$. $\frac{1}{2}(x-\frac{\sin (2x)}{2})+C$.

Step 12.2:

Distribute the $\frac{1}{2}$. $\frac{x}{2}-\frac{1}{2}(\frac{\sin (2x)}{2})+C$.

Step 12.3:

Simplify the multiplication. $\frac{x}{2}-\frac{\sin (2x)}{4}+C$.

Step 12.4:

Final expression. $\frac{x}{2}-\frac{\sin (2x)}{4}+C$.

Step 13:

Write the result in a standard form. $\frac{1}{2}x-\frac{1}{4}\sin (2x)+C$.

Knowledge Notes:

1. Half-Angle Formula: The half-angle formulas are trigonometric identities that express trigonometric functions of half angles in terms of the full angle. For sine, the formula is ${\sin }^2(x)=\frac{1-\cos (2x)}{2}$.

2. Integration by Substitution (u-Substitution): This technique involves changing the variable of integration to simplify the integral. It is the reverse process of the chain rule in differentiation.

3. Power Rule for Integration: The power rule states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\ne -1$.

4. Constant Multiple Rule in Integration: If $k$ is a constant and $f(x)$ is a function, then $\int kf(x)dx=k\int f(x)dx$.

5. Integration of Trigonometric Functions: The integral of $\cos (x)$ with respect to $x$ is $\sin (x)+C$, and the integral of $\sin (x)$ with respect to $x$ is $-\cos (x)+C$.

6. Simplification: The process of combining like terms and applying arithmetic operations to make an expression easier to understand or work with.

7. Substitution Back: After integrating with a substitution, it is necessary to substitute back the original variable to express the antiderivative in terms of the original variable.