Evaluate the line integral with respect to arc length.3 variables

Question

Evaluate the line integral with respect to arc length where C is given by ∫_c x ds

x = t, y = t sin t, z = t cos t, 0 ≤ t ≤ √(2)

firstly subbed t into equation as x=t

then differentiated x, y and z:

x'=1

y'=sint + tcost

z'=cost - tsint

these should then be subbed into the formula like this:

√2

∫ t*√((1)² + (sint+tcost)²+(cost-tsint)²) dt

0

Im now stuck as im unsure of how to progress from this point, have tried simplifying with trig identities but no luck.

Answer 1

∫₀^√2t√[1² + (sint + tcost)² + (cost - tsint)²]dt = ∫₀^√2t√(2 + t²)dt = 1/2·2/3(2 + t²)^3/2₀^√2= 1/3(8 - 2√2)= 2(4 - √2)/3 ≈

1.724