Integrate Using u-Substitution integral of e^(cos(x))sin(x) with respect to x
The question asks for the evaluation of a definite or indefinite integral (it is not specified which) using a method of integration called u-substitution. The function to be integrated is e^(cos(x)) * sin(x), where e represents the mathematical constant approximately equal to 2.71828, cos(x) is the cosine function, and sin(x) is the sine function. U-substitution involves changing the variable of integration from x to a new variable u, which is a function of x, in this case likely u = cos(x), to simplify the integral and make it easier to solve.
$\int e^{cos \left(\right. x \left.\right)} sin \left(\right. x \left.\right) d x$
Answer
Solution:
Step 1
Assign $u = \cos(x)$. Consequently, $du = -\sin(x)dx$, which implies $-\frac{1}{\sin(x)}du = dx$. Substitute $u$ and $du$ into the integral.
Step 1.1
Set $u = \cos(x)$ and calculate $\frac{du}{dx}$.
Step 1.1.1
Compute the derivative of $\cos(x)$: $\frac{d}{dx}[\cos(x)]$.
Step 1.1.2
The rate of change of $\cos(x)$ with respect to $x$ is $-\sin(x)$: $-\sin(x)$.
Step 1.2
Express the integral in terms of $u$ and $du$: $\int -e^u du$.
Step 2
Extract the constant factor $-1$ from the integral: $-\int e^u du$.
Step 3
Integrate $e^u$ with respect to $u$: $-(e^u + C)$.
Step 4
Condense the expression: $-e^u + C$.
Step 5
Substitute $u$ back with $\cos(x)$: $-e^{\cos(x)} + C$.
Knowledge Notes:
u-Substitution: This is a technique used in integration that involves substituting a part of the integrand with a new variable $u$. This simplifies the integral, making it easier to solve.
Derivative of $\cos(x)$: The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$. This is a fundamental result in calculus and is derived from the limit definition of a derivative.
Integration of $e^u$: The integral of $e^u$ with respect to $u$ is $e^u + C$, where $C$ is the constant of integration. This is because the rate of change of the exponential function is proportional to the function itself.
Constant Factor in Integration: A constant factor can be moved outside the integral sign. This property simplifies the integration process by allowing us to integrate the function without the constant and then multiply the result by the constant.
Back-Substitution: After integrating with respect to $u$, we need to substitute back to the original variable to express the antiderivative in terms of the original variable. This step completes the u-substitution process.
