Anirudh's Apostol Calculus Problems thread

[WiFO215]

Some time back I made a thread saying I would be buying a calculus book. I ended up buying Apostol as the Indian edition is only 6$. I have made a thread about this because I can see that there are going to be lots of problems I cannot do and I haven't a solution manual.

1.$$\int$$ (a² - x²)ⁿdx



2. I am supposed to use integration by parts to do this sum.



3. Using integration by parts,

x(a² - x²)ⁿ + n$$\int$$2x²(a² - x²)^n-1

But he gets a different answer which is

[x(a² - x²)ⁿ + 2a²n$$\int$$(a² - x²)^n-1 dx ]/ 2n+1 + C

 

[HallsofIvy]

HOW did you do the integration by parts? What did you use for u and dv?

 

[WiFO215]

U = (a²- x²)ⁿ
V = x

I missed an n in the original equation after the integral. OK. Now what?

 

[WiFO215]

OK. I worked backwards and solved it. But keep this thread open. I shall post any other exercises I have problems with.

 

[WiFO215]

A function f, continuous on the positive real axis, has the property that for all choices of x>0 and y>0, the integral
$$\int$$$$^{xy}_{x}$$f(t)dt
is independent of x (and therefore depends only on y). If f(2) = 2, compute value of integral A(x) = $$\int$$$$^{x}_{1}$$f(t)dt for all x>0