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 $\frac{1 - \cos(2x)}{2}$.
$\int \frac{1 - \cos(2x)}{2} \, dx$
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.
$\frac{1}{2} \left( \int dx - \int \cos(2x) \, dx \right)$
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 $\frac{d}{dx}(x^n) = nx^{n-1}$ 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.
$\frac{1}{2} \left( x + C - \frac{1}{2} \int \cos(u) \, du \right)$
Step 9:
Integrate $\cos(u)$ with respect to $u$.
$\frac{1}{2} \left( x + C - \frac{1}{2} (\sin(u) + C) \right)$
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:
Half-Angle Formula: This trigonometric identity is used to simplify the integral of $\sin^2(x)$. The half-angle formula for sine is $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.
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.
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 $\sin(x)$.
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 $du = g'(x)dx$.
Power Rule for Differentiation: This rule is used to find the derivative of $x^n$, which is $nx^{n-1}$. In the case of $u = 2x$, the derivative with respect to $x$ is $2$.
Simplification and Rearrangement: After integration, expressions are simplified and constants are combined to provide the final result.
