Can you please show me the process that determines \int_{-1}^1

b2sonicxh

b2sonicxh

Answered question

2022-01-03

Can you please show me the process that determines 11dxx2+1=π? Unfortunately I only know that the derivative of sin1(x) is equal to the integrand of that, but I don't actually know what to do with it.

Answer & Explanation

Kirsten Davis

Kirsten Davis

Beginner2022-01-04Added 27 answers

You are right in saying that indeed, 11x2dx=arcsin(x). Now, you need to use the Fundamental Theorem of Calculus, which says that
abf(x)dx=F(b)F(a)
where F is a function satisfying
F′(x)=f(x)
Alex Sheppard

Alex Sheppard

Beginner2022-01-05Added 36 answers

Because you know that
ddxsin1x=1x2+1
you are already most of the way there. This tells you that
11dxx2+1=sin1(1)sin1(1)=π2(π2)=π
This is due to the Fundamental Theorem of Calculus, which says that for a continuous function f on [a,b], with antiderivative F, then
abf(x)dx=F(b)F(a)
karton

karton

Expert2022-01-11Added 613 answers

Denote x=sin(t)Because 1x1 then π2tπ2. We have:I=111x2+1dx=π2π21(sin(t))2+1dsin(t)=π2π2costcostdt=π2π21dtI=π

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