an equivalent representation of the definite integral of 3 to 1 2xcos(x^2)dx is
Let u = x^2
then we have integral from 9 to 1 of cos(u) du
Oh, you want me to transform the definite integral into an equivalent expression. Okay, let me put on my math clown hat for a moment. *Puts on a clown hat* Ta-da!
The equivalent representation of the definite integral of 3 to 1 of 2xcos(x^2)dx is:
∫[1 to 3] 2xcos(x^2)dx
Alright, that's the serious version. But if you want a clown-style equivalent representation, here it goes:
Imagine you're at the circus, and you have a clown juggling two things - an "x" and a "cos(x^2)" function, because why not? Now, put a big integral sign in front and let the clown juggle from x = 1 to x = 3, because even clowns need limits. And voila, you have your equivalent representation! Enjoy the circus of calculus!
To find an equivalent representation of the definite integral of 3 to 1 of 2x*cos(x^2)dx, we can use the substitution method.
Let's substitute u = x^2. Taking the derivative of both sides with respect to x, we get du = 2x*dx. Rearranging, we have dx = du/(2x).
Now, let's substitute these values into the original integral:
∫(3 to 1) 2x*cos(x^2)dx = ∫(3 to 1) 2x*cos(u)*(du/(2x))
The x and 2x in the numerator cancel out:
∫(3 to 1) cos(u) du
Since the limits of integration do not change, the equivalent representation of the definite integral is:
∫(3 to 1) cos(u) du, where u = x^2.
To find an equivalent representation of the definite integral of 3 to 1 of 2xcos(x^2)dx, we can use properties of definite integrals along with the Fundamental Theorem of Calculus.
First, let's start with the general form of the definite integral:
∫[a to b] f(x) dx
In our case, a = 3 and b = 1, and f(x) = 2xcos(x^2). So we have:
∫[3 to 1] 2xcos(x^2) dx
To find an equivalent representation, we can apply the substitution method. Let's set u = x^2, then du = 2x dx. Rearranging, we have dx = du / (2x).
Substituting this into our integral, we get:
∫[3 to 1] 2xcos(x^2) dx = ∫[3 to 1] cos(u) (du/(2x))
Now, we can split this integral into two parts:
∫[3 to 1] cos(u) du - ∫[3 to 1] cos(u)/(2x) du
The first integral, ∫[3 to 1] cos(u) du, is simpler to evaluate. It is the antiderivative of cos(u) with respect to u, which is sin(u):
∫[3 to 1] cos(u) du = sin(u) | [3 to 1] = sin(1) - sin(3)
For the second integral, we can pull out the constant 1/2 and write it as:
(1/2) ∫[3 to 1] cos(u)/x du
Here, we can see that x is constant with respect to u, so we can pull it out of the integral:
(1/2x) ∫[3 to 1] cos(u) du
Evaluating this integral, again using the antiderivative of cos(u) which is sin(u), we get:
(1/2x) [sin(u)] | [3 to 1] = (1/2x)(sin(1) - sin(3))
Therefore, an equivalent representation of the given definite integral of 3 to 1 of 2xcos(x^2)dx is:
sin(1) - sin(3) - (1/2x)(sin(1) - sin(3))
where x is a variable.