Calc 2: Suppose, for b>a>0, int[0,b] e^x dx = 2 int[0,a] e^x dx....

[victoriaaa29]

Question 1. Let b > a > 0 be two positive numbers. Suppose that

. . . . .\(\displaystyle \displaystyle \int_0^b\, e^x\, dx\, =\, 2\, \int_0^a\, e^x\, dx\)

Express b in terms of a.

Question 2. Find a function f (x) and a number a such that, for all x > 0,

. . . . .\(\displaystyle \displaystyle 16 + \int_a^{\sqrt{\strut x\,}}\, \dfrac{f(t)}{t^2}\, dt\, =\, 2\sqrt{\strut x\,}\)

Question 3. Let a, b > 0 be two positive numbers. Show that

. . . . .\(\displaystyle \displaystyle \int_0^1\, x^a\, (1\, -\, x)^b\, dx\, =\, \int_0^1\, x^b\, (1\, -\, x)^a\, dx\)

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I don't even know where to begin with these three problems, any help greatly appreciated, thanks!

 

[MarkFL]

1.) Let:

\(\displaystyle u=e^x\implies du=e^x\,dx\)

And the equation becomes:

\(\displaystyle \displaystyle \int_1^{e^b}\,du=2\int_1^{e^a}\,du\)

Can you proceed?

2.) Try computing:

\(\displaystyle \displaystyle \frac{d}{dx}\left(16+\int_a^{\sqrt{x}} \frac{f(t)}{t^2}\,dt\right)= \frac{d}{dx}\left(2\sqrt{x}\right)\)

What do you get?

3.) Use the fact that in a definite integral, the variable of integration is a "dummy" variable (i.e. it gets integrated out), which means:

\(\displaystyle \displaystyle \int_a^b f(u)\,du=\int_a^b f(v)\,dv\)

Can you now use an appropriate substitution on either side of the equation?

 

[victoriaaa29]

Thank you so much for the start! Makes more sense now! Thank you!!!