Integrate Using u-Substitution integral of (4sin(x))/(3+cos(x)) with respect to x

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:

1. Choosing an appropriate substitution for $u$.

2. Differentiating $u$ with respect to $x$ to find $du$.

3. Rewriting the integral in terms of $u$ and $du$.

4. Integrating with respect to $u$.

5. Simplifying the result if necessary.

6. 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.