Definite Integral of Product/Composite Function Given Graph

[JessTheMess]

Homework Statement

Given the graph of f(x) shown below, find the value of the integral.
Photo attached.

Homework Equations

[/B]
∫²₃ 5x·f(x²)dx

The Attempt at a Solution

[/B]
I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x²)) doesn't make much sense to me without knowing the actual function that represents f(x).

I also tried using the equation of the semi-circle to find it, but the integration got too complicated.

Does anyone have a hint on how to start the problem and/or deal with the composite function in the integral? Thank you!

 

[eys_physics]

I would do the change of variables
$$t=x^2$$.

 

[JessTheMess]

When I do u-substitution, I also run into a problem.

t=x^2
dt = 2xdx
dx = dt/2x

∫5xf(t)dt/2x

5/2 ∫f(t)dt

How do I take the integral of f(t) when I don’t know what function f(t) is?

 

[PeroK]

How do I take the integral of f(t) when I don’t know what function f(t) is?

I think you are supposed to work out what function $f$ is from the graph.

 

[Ray Vickson]

Homework Statement

Given the graph of f(x) shown below, find the value of the integral.
Photo attached.

Homework Equations

[/B]
∫²₃ 5x·f(x²)dx

The Attempt at a Solution

[/B]
I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x²)) doesn't make much sense to me without knowing the actual function that represents f(x).

I also tried using the equation of the semi-circle to find it, but the integration got too complicated.

Does anyone have a hint on how to start the problem and/or deal with the composite function in the integral? Thank you!

If one takes the most "obvious" form of the function $f(x)$, the integral $\int_3^2 5x f(x) \, dx$ is negative; I hope you see why. However, if your integral, instead, really is $\int_3^2 5x f(x^2) \, dx$ (exactly as written) the integral is a pure imaginary number with a negative imaginary part.