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 $\frac{1-\cos (2x)}{2}$.
$\int \frac{1-\cos (2x)}{2}dx$

Step 2:

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

Step 3:

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

Step 4:

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

Step 5:

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

Step 6:

Let $u=2x$ and calculate $du$.

Step 6.1:

Define $u$ as $2x$. 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, which states $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for $n=1$.
$2\cdot 1$

Step 6.1.4:

Multiply $2$ by $1$ to find $du$.
$2$

Step 6.2:

Substitute $u$ and $\frac{1}{2}du$ into the integral.
$\frac{1}{2}(x+C-\int \cos (u)\frac{1}{2}du)$

Step 7:

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

Step 8:

Factor out 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:

Replace $u$ with $2x$.
$\frac{1}{2}(x-\frac{1}{2}\sin (2x))+C$

Step 12:

Simplify the result.

Step 12.1:

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

Step 12.2:

Apply the distributive property.
$\frac{1}{2}x-\frac{1}{2}\left(\frac{\sin (2x)}{2}\right)+C$

Step 12.3:

Combine $\frac{1}{2}$ and $x$.
$\frac{x}{2}-\frac{1}{2}\left(\frac{\sin (2x)}{2}\right)+C$

Step 12.4:

Multiply $\frac{1}{2}$ by $\frac{\sin (2x)}{2}$.

Step 12.4.1:

Multiply $\frac{1}{2}$ by $\frac{\sin (2x)}{2}$.
$\frac{x}{2}-\frac{\sin (2x)}{4}+C$

Step 12.4.2:

Multiply $2$ by $2$.
$\frac{x}{2}-\frac{\sin (2x)}{4}+C$

Step 13:

Reorder the terms for the final answer.
$\frac{1}{2}x-\frac{1}{4}\sin (2x)+C$

Knowledge Notes:

To solve the integral of ${\sin }^2(x)$ using u-substitution, we employ several mathematical concepts and techniques:

1. Half-Angle Formula: This trigonometric identity is used to simplify the integral of ${\sin }^2(x)$. The half-angle formula for sine is ${\sin }^2(x)=\frac{1-\cos (2x)}{2}$.

2. Constant Multiple Rule: This rule allows us to pull constants out of an integral. It states that $\int k\cdot f(x)dx=k\cdot \int f(x)dx$ where $k$ is a constant.

3. Integration of Basic Functions: The integral of $1$ with respect to $x$ is $x$, and the integral of $\cos (x)$ with respect to $x$ is $\sin (x)$.

4. U-Substitution: This technique is used for integrating composite functions and involves a change of variables to simplify the integral. If $u=g(x)$, then $du=g^{\prime }(x)dx$.

5. Power Rule for Differentiation: This rule is used to find the derivative of $x^n$, which is $n{x}^{n-1}$. In the case of $u=2x$, the derivative with respect to $x$ is $2$.

6. Simplification and Rearrangement: After integration, expressions are simplified and constants are combined to provide the final result.