Solved Examples of Differentiation and Integration Formula
Example 1: Differentiate [Tex]\bold{y= \frac{1}{3x+1}} [/Tex] with respect to x.
Solution:
Let [Tex]y = \frac{1}{3x+1} [/Tex]
⇒ [Tex]\frac{dy}{dx} = \frac{d}{dx}(\frac{1}{3x+1}) [/Tex]
⇒ [Tex]\frac{dy}{dx} = -\frac{3}{(3x+1)^2}[/Tex]
Example 2: Differentiate the following: i) x3 ii)[Tex] \bold{\frac{1}{x^3+1}} [/Tex][Tex] \frac{1}{x^3+1}[/Tex]
Solution:
i) Let y = x3
⇒\frac{dy}{dx} = \frac{d}{dx}(x^3)
⇒[Tex]\frac{dy}{dx} = 3x^2 [/Tex]
ii) Let[Tex] y = \frac{1}{x^3+1} [/Tex]
Using, Quotient Rule,
[Tex]\frac{dy}{dx} = \frac{-3x^2}{(x^3+1)^2}[/Tex]
Example 3: Find of derivative of [Tex]\bold{y = \frac{e^x-e^{-x}}{e^{-x}+e^x}} [/Tex] With respect to x.
Solution:
Let [Tex]y = \frac{e^x-e^{-x}}{e^{-x}+e^x} [/Tex]
⇒ [Tex]\frac{dy}{dx} =\frac{(e^{-x}+e^x)(e^x+e^{-x}) – (e^{-x}+e^x)(e^x+ e^{-x})}{(e^{-x}+e^x)^2} [/Tex]
⇒ [Tex]\frac{dy}{dx} = \frac{e^{-2x}+e^{2x} -e^{-2x}-e^{2x}}{(e^{-x}+e^x)^2} [/Tex]
⇒ [Tex]\frac{dy}{dx} = 0[/Tex]
Example 4: Differentiate [Tex]\bold{y = \frac{e^x-e^{-x}}{e^{-x}+e^x}} [/Tex] with respect to x.
Solution:
[Tex]y = \frac{e^x-e^{-x}}{e^{x}+e^{-x}}= \frac{e^{2x}-1}{e^{2x}+1} [/Tex]
⇒[Tex] \frac{dy}{dx}=\frac{\frac{d}{dx}(e^{2x}-1)(e^{2x}+1)-\frac{d}{dx}(e^{2x}+1)(e^{2x}-1)}{(e^{2x}+1)^2}[/Tex]
⇒ [Tex]dy/dx = \frac{2e^{2x}(e^{2x}+1)-2e^{2x}(e^{2x}+1)}{(e^{2x}+1)^2}[/Tex]
⇒ [Tex]dy/dx = \frac{4e^{2x}}{(e^{2x}+1)^2} [/Tex]
Example 5: Differentiate y = Sec2x with respect to x.
Solution:
Let y = sec2x
⇒ [Tex]\frac{dy}{dx} = 2secx(secx tanx) [/Tex]
⇒ [Tex]\frac{dy}{dx} [/Tex] = 2 sec2 x tan x
Example 6: Differentiate sec2x + cos2x.
Solution:
y = sec2x + cos2x
⇒ [Tex]\frac{dy}{dx} = 2sinx cosx + (-2sin2x) [/Tex]
⇒ [Tex]\frac{dy}{dx} = 2sinx cosx -2sin2x[/Tex]
Example 7: Integrate √x with respect to x.
Solution:
y = ∫√x dx
⇒ y = [Tex]∫x^{\frac{1}{2}} dx [/Tex]
⇒ [Tex]y = \frac{ x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + c [/Tex]
⇒ [Tex]y = \frac{2}{3}x^{\frac{3}{2}} + c[/Tex]
Example 8: Integrate the following:
(i) e2x (ii) eax
Solution:
i) y=∫e2x
⇒ [Tex]y = \frac{e^{2x}}{2} + c [/Tex]
ii) y=∫eax
⇒ [Tex]y = \frac{e^{ax}}{a} + c [/Tex]
Example 9: Integrate sin2x+ cos2x.
Solution:
y = ∫(sin2x + cos2x)dx
⇒ y = ∫dx
⇒ y = x + c
Example 10: Integrate sin 2x + cos 2x.
Solution:
y = ∫(sin2x + cos2x)dx
⇒ y = ∫sin2xdx + ∫cos2x dx
⇒ y = [Tex]\frac{-cos2x}{2}+ \frac{sin2x}{2}+ c [/Tex]
⇒ [Tex]y = \frac{1}{2}(sin2x -cos2x) + c [/Tex]
Example 11: Find the area bounded by the curve y = Sinx between x= 0 and x = 2π.
Solution:
Let y = Sinx
The graph of y = sinx is like,
Required area = Area of OABO + Area of BCDB
⇒ Required area [Tex]= \int_{0}^{π}|sinx|dx + \int_{π}^{2π}|sinx|dx[/Tex]
⇒ Required area [Tex]= \int_{0}^{π}sinx dx + \int_{π}^{2π}-sinxdx[/Tex]
⇒ Required area [Tex]= \left[ -cosx \right]_{0}^{π} + \left[ cosx \right]_{π}^{2π}[/Tex]
⇒ Required area = -cosπ + cos0 + cos2π- cosπ
⇒ Required area = 4 sq units.
Example 12: The area bounded by the region of curve y2 = x and the lines x = 1, x= 4, and the x-axis is :
Solution:
Let y2 = x a curve region bounded by the lines x = 1 and x = 4 about x-axis.
Required Area (Shaded Area) =[Tex] \int_{1}^{4}|y|dx [/Tex]
⇒ Required area [Tex]= \int_{1}^{4}\sqrt{x}dx[/Tex]
⇒ Required area [Tex]= \left[\frac{ x^{\frac{3}{2}}}{\frac{3}{2}} \right] [/Tex]
⇒ Required area [Tex]= \frac{2}{3}\left[ {4}^{\frac{3}{2}} – 1 \right][/Tex]
⇒ Required area [Tex]= \frac{14}{3} sq. units. [/Tex]
Example 13: The area of the region area, integrate x w.r.t. y and take y = 2 as the lower limit and y = 4 as the upper limit. The given curve x^2 = 4y is a parabola, which is symmetrical about the y-axis.
Solution:
The given curve is parabola x2 = 4y which is symmetric to the y-axis.
The area bounded by the curve is shaded portion of the graph.
Required Area = [Tex]\int_{2}^{4}|x|dy[/Tex]
⇒ Required area [Tex]= \int_{2}^{4}2\sqrt{y}dy [/Tex]
⇒ Required area [Tex]= 2\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}} \right]_{2}^{4}[/Tex]
⇒ Required area [Tex]= \frac{8}{3}\left[ 4 -\sqrt{2} \right] sq. units.[/Tex]
Differentiation and Integration Formula
Differentiation and Integration are two mathematical operations used to find change in a function or a quantity with respect to another quantity instantaneously and over a period, respectively. Differentiation is an instantaneous rate of change and it breaks down the function for that instant with respect to a particular quantity while Integration is the average rate of change that causes the summation of continuous data of a function over the given period or range. Both are inverse of each other.
In this article, we will learn about what is differentiation, what is integration, and the formulas related to Differentiation and Integration.
Table of Content
- What is Differentiation?
- How to Differentiate a Function
- Differentiation Formulas
- Derivative of Algebraic Functions
- Derivative of Exponential Functions
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Differentiation by Parts
- What is Integration?
- How to Integrate Function
- Integration Formulas
- Integration of Algebraic Functions
- Integration of Exponential Functions
- Integration of Trigonometric Functions
- Integration By Parts
- Area Under the Curve
- Differentiation and Integration Formulas
- Properties of Differentiation and Integration
- Difference between Differentiation and Integration
- Solved Examples of Differentiation and Integration Formula
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