Integrate Using u-Substitution integral of (4sin(x))/(3+cos(x)) with respect to x
The question asks for the evaluation of a given integral, specifically using the method of u-substitution. The integral in question is of the form ∫(4sin(x))/(3+cos(x)) dx, where the goal is to find the antiderivative of the function (4sin(x))/(3+cos(x)) with respect to the variable x. U-substitution is a technique often used in calculus to make complex integrals more manageable by substituting a part of the integral with a new variable u, which then simplifies the integral into a form that is easier to integrate. The task includes determining an appropriate substitution that will simplify the integral and performing the necessary algebraic manipulations to arrive at the answer.
$\int \frac{4 sin \left(\right. x \left.\right)}{3 + cos \left(\right. x \left.\right)} d x$
Answer
Solution:
Step 1: Define a new variable u
Let $u = 3 + \cos(x)$. This implies that $du = -\sin(x)dx$, which can be rearranged to $-\frac{1}{\sin(x)}du = dx$. We will express the integral in terms of $u$ and $du$.
Step 1.1: Compute the derivative of u with respect to x
Calculate $\frac{du}{dx}$ for $u = 3 + \cos(x)$.
Step 1.1.1: Apply differentiation to the expression
Take the derivative of $3 + \cos(x)$ with respect to $x$: $\frac{d}{dx}(3 + \cos(x))$.
Step 1.1.2: Apply the Sum Rule in differentiation
The derivative of a sum is the sum of the derivatives: $\frac{d}{dx}(3) + \frac{d}{dx}(\cos(x))$.
Step 1.1.2.1: Differentiate the constant term
The derivative of a constant is zero: $0 + \frac{d}{dx}(\cos(x))$.
Step 1.1.3: Differentiate the cosine function
The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$: $0 - \sin(x)$.
Step 1.1.4: Combine the terms
Combine the derivatives: $-\sin(x)$.
Step 1.2: Rewrite the integral in terms of u and du
Substitute the expressions for $dx$ and $u$ into the integral: $\int \frac{-4}{u} du$.
Step 2: Simplify the integrand
Break down the fraction in the integral: $\int -\frac{4}{u} du$.
Step 3: Factor out constants from the integral
Extract the constant $-1$ from the integral: $- \int \frac{4}{u} du$.
Step 4: Factor out the constant 4
Extract the constant $4$ from the integral: $-4 \int \frac{1}{u} du$.
Step 5: Combine the constants
Multiply the constants $-1$ and $4$: $-4 \int \frac{1}{u} du$.
Step 6: Integrate with respect to u
Integrate $\frac{1}{u}$ with respect to $u$: $-4(\ln|u| + C)$.
Step 7: Simplify the expression
Simplify the result: $-4\ln|u| + C$.
Step 8: Back-substitute the original variable
Replace $u$ with $3 + \cos(x)$: $-4\ln|3 + \cos(x)| + C$.
Knowledge Notes:
In this problem, we are asked to integrate the function $\frac{4\sin(x)}{3+\cos(x)}$ with respect to $x$ using the method of u-substitution. The u-substitution method is a technique used in calculus to simplify the integration process by substituting a part of the integrand with a new variable $u$. This often simplifies the integral into a form that is easier to integrate.
The steps involved in this process include:
Choosing an appropriate substitution for $u$.
Differentiating $u$ with respect to $x$ to find $du$.
Rewriting the integral in terms of $u$ and $du$.
Integrating with respect to $u$.
Simplifying the result if necessary.
Substituting back the original variable to express the result in terms of $x$.
In this case, we chose $u = 3 + \cos(x)$ and found that $du = -\sin(x)dx$. After rewriting the integral in terms of $u$, we integrated and then substituted back to get the final result in terms of $x$. The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u|$, and we must remember to include the constant of integration $C$ in our final answer. The absolute value in the logarithm ensures that the argument is always positive, as the logarithm of a negative number is not defined in the real number system.
