To determine

To evaluate:

The given definite integral ∫01cos⁡πt2dt.

Answer

2π

Explanation

1) Concept:

i) The substitution rule for definite integral:

If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then ∫abfgxg'(x)dx=∫g(a)g(b)f(u)du.

ii)

∫abcfxdx=c∫abfxdx

2) Given:

∫01cos⁡πt2dt

3) Calculation:

The given integral is

∫01cos⁡πt2dt. Here, use the substitution method.

Substitute πt2=u. Solving for t

 t=2π u

Differentiating with respect to t

 dt=2πdu

The limits changes, the new limits of integration are calculated by substituting

when t=0,  u=0   

when t=1,  u=π/2

Therefore, the given integral becomes

∫01cos⁡(πt/2)dt=∫0π2cos⁡u2πdu

=2π∫0π2cosudu

Integrating this, we get

=2π(sinu)0π/2

=2πsin⁡π2-sin0

=2π(1-0)

=2π

Conclusion:

Therefore,

∫01cos⁡(πt/2)dt=2π