f(x) = e^(sin(x))cos(x)
f(x) = e^(sin(x))cos(x)
I=∫f(x)
= ∫e^(sin(x))cos(x)dx
use substitution u=sin(x),
du=cos(x)dx
I=∫e^u du
=e^u +C'
=e^(sin(x))+C
The given function is f(x) = e^(sin(x))cos(x).
To understand this function, we need to break it down into its components:
1. e^(sin(x)): The exponential function e^(sin(x)) represents Euler's number (approximately 2.71828) raised to the power of the sine of x. The sine function ranges from -1 to 1, so e^(sin(x)) can also take on a wide range of values.
2. cos(x): The cosine function cos(x) gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle with an angle x. It oscillates between -1 and 1 as x varies.
By multiplying e^(sin(x)) and cos(x) together, we obtain the function f(x) = e^(sin(x))cos(x). This function combines the exponential growth of e^(sin(x)) with the oscillatory nature of cos(x), resulting in a function that can have highly intricate behavior.
To analyze this function further, we can graph it or evaluate it at specific values of x. Additionally, we can apply calculus techniques like finding the derivative or integral of the function if we want to explore its rate of change or area under the curve.