Evaluate the intergral

Question

Consider the integral of ranges from 0 to pi/4 sinx/x (a=0, b=pi/4 function is sinx/x)use teh fact that sinx=x-x^3/3! near a=0 to evaluate the integral

Answer 1

π/4

∫ sin(x)/x dx

0

This is not an integral that can be solved using elementary techniques, which is why they have us use a Taylor series approximation to evaluate the integral.

Using the fact that sin(x) ≈ x - (x³/3!) for values close to 0, we get

π/4

∫ [x - (x³/6)]/x dx

0

π/4

∫ 1 - (x²/6) dx

0

(x - (x³/18)) evaluated from 0 to π/4

(π/4 - ((π/4)³/18)) - (0 - ((0)³/18))

π/4 - π³/1152

This can either be simplified to (288π-π³)/1152, or the decimal 0.758482992668 and rounded to the desired decimal.

Extra: For those interested, the answer to the integral of sin(x)/x actually has a defined function for it, naturally called the sine integral, Si(x), where x refers to the upper bound of the integral, the lower bound always being 0. So the exact answer to the integral here would be Si(π/4) = 0.758975881069, only half a thousandth away from our approximation!