Test: Improper Integrals - Civil Engineering (CE) MCQ

Integration of function is same as the Joining many small entities to create a large entity.

Add constant automatically
Let f(x) = e^x Sin(x)
∫ e^x Sin(x)dx = -e^x Cos(x) + ∫ e^x Cos(x)dx
∫ e^x Sin(x)d - ∫ e^x Cos(x)dx = ∫ e^x [Sin(x) - Cos(x)]dx
= -e^x Cos(x).

Since slope of a function is given by ^dy⁄_dx at that point. Hence, when ^dy⁄_dx = 0 means slope of a function is zero i.e, parallel to x axis.
Function is not a constant function since it has finite value at other points.

Integration of function y=f(x) from limit x1 < x < x₂ , y₁ < y < y₂, gives area of f(x) within x1 < x < x₂.

Add constant automatically
Given, e^t = x ⇒ dx = e^t dt,
Given,

Let, z = ln⁡(x) ⇒ dz = dxx
⇒ 

Add constant automatically
Given, 

= 

Let f(x) = e^x Sin(x)

∫ e^x Sin(x)dx = e^x Sin(x) – ∫ e^x Cos(x)dx

∫ e^x Sin(x)dx + ∫ e^x Cos(x)dx = ∫ e^x [Cos(x) + Sin(x)]dx

= e^x Sin(x).

Add constant automatically
Given,f(x) = 

= 

= 

= 

Since slope of a function is given by dy⁄dx at that point.Hence,when dy⁄dx = ∞ means slope of a function is 90 degree i.e,parallel to y axis.

Add constant automatically
Given, 

Let, z = ln⁡(x)

⇒ 

⇒