If f'(x) = 3x² - 4x + 5 and f(1) = 3, then find f(x).

A train started from Madurai Junction towards Coimbatore at 3 pm (time t = 0) with velocity v(t) = 20t + 50 kilometre per hour, where t is measured in hours. Find the distance covered by the train at 5 pm.

A tree is growing so that, after t - years its height is increasing at a rate of \({18\over \sqrt{t}}\) cm per year Assume that when t = 0, the height is 5 cm.
(i) Find the height of the tree after 4 years.
(ii) After how many years will the height be 149 cm?

At a particular moment, a student needs to stop his speedy bike to avoid a collision with the barrier ahead at a distance 40 metres away from him. Immediately he slows (retardation) the bike under braking at a rate of 8 metre/second². If the bike is moving at a speed of 24m/s, when the brakes are applied, would it stop before collision?

Evaluate : \(\int {1\over \sqrt{x+1}+\sqrt{x}}dx\)

Evaluate \(\int {x+3\over (x+2)^2(x+1)}dx\)

Integrate the following functions with respect to x : \(x+1\over (x+2)(x+3)\)

Integrate the following functions with respect to x : \({3x-9\over (x-1)(x+2)(x^2+1)}\)

Integrate the following with respect to x : \({2x-3\over x^2+4x-12}\)