Integrate Using u-Substitution integral of sin(x)^2 with respect to x
The problem asks to perform an integration using the u-substitution method on the function sin(x)^2 with respect to x. U-substitution is a technique often employed in calculus to simplify an integral by substituting a part of the integrand with a single variable (u), which hopefully makes the integral easier to solve. The question is essentially about finding the antiderivative of the square of the sine function using this method.
$\int \left(sin\right)^{2} \left(\right. x \left.\right) d x$
Answer
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 $nx^{n-1}$ 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:
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 $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.
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.
Power Rule for Integration: The power rule states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$.
Constant Multiple Rule in Integration: If $k$ is a constant and $f(x)$ is a function, then $\int k f(x) dx = k \int f(x) dx$.
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 $-\cos(x) + C$.
Simplification: The process of combining like terms and applying arithmetic operations to make an expression easier to understand or work with.
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.
