Integrate Using u-Substitution integral of (sin(x))/(1+cos(x)^2) with respect to x
In this calculus problem, you are asked to perform integration of the given function (sin(x)) / (1+cos(x)^2) with respect to the variable x. Specifically, the question directs you to use the technique of u-substitution to find the antiderivative. U-substitution is a method often used to simplify integrals by substituting part of the integrand with a new variable 'u', which turns the original integral into a simpler form that is easier to integrate. The challenge in such a problem lies in identifying the appropriate substitution that simplifies the integral effectively.
$\int \frac{sin \left(\right. x \left.\right)}{1 + \left(cos\right)^{2} \left(\right. x \left.\right)} d x$
Answer
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:
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.
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.
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.
Constants in Integration: Constants can be factored out of the integral, which simplifies the integration process.
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.
