From the Power Series table for trigonometric function, we have:
arctan(x) =sum_(n=0)^oo (-1)^n x^(2n+1)/(2n+1)
= x -x^3/3 +x^5/5 - x^7/7 + x^9/9-...
Applying it on the integral int_0^(1/2) arctan(x)/x dx where the integrand is f(x)=arctan(x)/x , we get:
int_0^(1/2) arctan(x)/x dx =int_0^1 arctan(x) *1/x dx
=int_0^(1/2)sum_(n=0)^oo (-1)^n x^(2n+1)/(2n+1) *1/xdx
=int_0^(1/2) sum_(n=0)^oo (-1)^n x^(2n+1)/(2n+1) *x^(-1)dx
=int_0^(1/2) sum_(n=0)^oo (-1)^n x^(2n+1-1)/(2n+1) dx
=int_0^(1/2) sum_(n=0)^oo (-1)^n x^(2n)/(2n+1) dx
=int_0^(1/2) [1 -x^2/3 +x^4/5 - x^6/7 +x^8/9-...] dx
Or
int_0^(1/2) arctan(x)/x dx =int_0^1 arctan(x) *1/x dx
=int_0^(1/2) 1/x* [x -x^3/3 +x^5/5 - x^7/7 +x^9/9-...] dx
= int_0^(1/2) [x/x -x^3/(3x) +x^5/(5x) - x^7/(7x) +x^9/(9x)-...] dx
=int_0^(1/2) [1 -x^2/3 +x^4/5 - x^6/7 +x^8/9-...] dx
To determine the indefinite integral, we integrate each term using Power rule for integration: int x^n dx = x^(n+1)/(n+1) .
int_0^(1/2) [1 -x^2/3+x^4/5 - x^6/7 +x^8/9-...] dx= [x -x^3/(3*3)+x^5/(5*5) - x^7/(7*7) +x^9/(9*9)-...]_0^1
=[x -x^3/9+x^5/25 - x^7/49 +x^9/81-...]_0^1
Apply definite integral formula: F(x)|a^b =F(b)-F(a) .
F(1/2) or F(0.5)=0.5 -0.5^3/9+0.5^5/25 - 0.5^7/49 +0.5^9/81-...
= 1/2 -1/72+1/800-1/6272+1/41472 -...
F(0)= 0 -0^3/9+0^5/25 - 0^7/49+0^9/81-...
=0-0+0-0+0-...
All the terms are 0 then F(0)=0 .
We can stop on the 5th term (1/41472 ~~2.4113x10^(-5)) since we only need an error less than 0.0001.
F(1/2)-F(0)= [1/2 -1/72+1/800-1/6272+1/41472]-[0]
=1/2 -1/72+1/800-1/6272+1/41472
=0.487225785
Then, the approximated integral value will be:
int_0^(1/2) arctan(x)/x dx~~0.4872
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