Integrate Using u-Substitution integral of (sin(x))/(1+cos(x)^2) with respect to x

Solution:

Step 1:

Choose $u=\cos (x)$. Consequently, we have $du=-\sin (x)dx$, which implies $-\sin (x)dx=du$. Now express the integral in terms of $u$ and $du$.

Step 1.1:

Define $u$ as $\cos (x)$. Compute the derivative $\frac{du}{dx}$.

Step 1.1.1:

Differentiate $\cos (x)$. $\frac{d}{dx}[\cos (x)]$

Step 1.1.2:

The derivative of $\cos (x)$ with respect to $x$ is $-\sin (x)$. Therefore, $du=-\sin (x)dx$.

Step 1.2:

Substitute $u$ and $du$ into the integral. $\int \frac{-1}{1+u^2}du$

Step 2:

Extract the negative sign from the integral. $\int -\frac{1}{1+u^2}du$

Step 3:

As $-1$ is a constant with respect to $u$, factor it out of the integral. $-\int \frac{1}{1+u^2}du$

Step 4:

Represent the number $1$ as $1^2$. $-\int \frac{1}{1^2+u^2}du$

Step 5:

The integral of $\frac{1}{1^2+u^2}$ with respect to $u$ is $\arctan (u)+C$. $-(\arctan (u)+C)$

Step 6:

Simplify the expression. $-\arctan (u)+C$

Step 7:

Substitute back the original variable, replacing $u$ with $\cos (x)$. $-\arctan (\cos (x))+C$

Knowledge Notes:

1. U-Substitution: A technique used in integration, which involves substituting part of the integrand with a new variable $u$. This simplifies the integral into a form that is easier to solve.

2. Derivative of Cosine: The derivative of $\cos (x)$ with respect to $x$ is $-\sin (x)$. This is a fundamental result from calculus and is used when applying u-substitution involving trigonometric functions.

3. Integral of $1/(1+u^2)$: The integral of $\frac{1}{1+u^2}$ with respect to $u$ is $\arctan (u)+C$, where $C$ is the constant of integration. This is a standard result from integral calculus.

4. Constants in Integration: Constants can be factored out of the integral, which simplifies the integration process.

5. Back-Substitution: After integrating with respect to $u$, the original variable (in this case, $x$) is substituted back into the expression to return to the original variable of integration.