Objective Questions

Given the function y = e^x, find f(x+h) - f(x).

Evaluate lim_x→0 (sin x) / x.

Find the area between the curve y = x³ - 6x² + 8x and the x-axis in square unit…

Obtain the integral ∫sin(2x) / sin(x) dx.

Find the first derivative of sec^-1(5x) with respect to x.

By Wallis' formula, evaluate ∫₀^π/2 cos⁵ x sin<…

Given that ∫₀^θ 4 sin(2x)cos⁴(2x) dx = 4/…

Given that ∫₀^θ 5 tan x sec⁴ x dx = 15/4,…

Evaluate ∫₀^π/4 sec² x e^{4 tan x} dx.

If f(x) = x e^-x, find f''(x).

Find two positive numbers whose sum is 105 and product is maximum.

Find the integral ∫(x+1) / (x² - x - 6) dx.

Evaluate d²/dx² (3√x - 5/(x√x)).

The maximum value of the function f(x) = x³ - 27x + 15 occurs at x = ?

Given that x = a sin t and y = a cos t, what is y' (dy/dx)?

Integrate ∫ (1 / (x - x ln x)) dx.

Integrate ∫ x eˣ dx.

Given the function y = eˣ, the expression f(x+h) - f(x) is:

Evaluate lim (sin x / x) as x → 0.

Obtain the integral ∫ (sin 2x / sin x) dx.

By Wallis' formula, evaluate ∫₀^(π/2) cos⁵ x sin² x dx.

Obtain the integral ∫ (eˣ / (4 + eˣ)) dx.

Obtain the integral ∫ sec x dx.

By Wallis' formula, evaluate ∫₀^(π/2) cos⁴ x dx.

If f(x) = x e⁻ˣ, find f''(x).

Given that x = a sin t, y = a cos t, find y' (dy/dx).

Evaluate ∫₁⁴ (1/8 x^(3/2) - 4/x) dx, for x > 0.

Given that ∫₀^θ 4 sin(2x)cos⁴(2x) dx = 4/5, where 0 < θ < π, find the exact val…

Given that ∫₀² kx² e^(x³) dx = (2/3) (e⁸ - 1), find the value of k.

Find the integral ∫ (x+1)/(x²-x-6) dx.