Indefinite Integrals Examples
Example 1: Find the integral for the given function f(x), f(x) = sin(x) + 1
Solution:
Given f(x) = sin(x) + 1
sin(x) is a standard function, and it’s anti-derivative is,
=∫ f(x)dx
= ∫ (sin(x) + 1)dx
= [Tex]\int sin(x)dx + \int 1dx[/Tex]
= [Tex]-cos(x) + x + C[/Tex]
Example 2: Find the integral for the given function f(x), f(x) = 2ex
Solution:
Given f(x) = 2ex
ex is a standard function, and it’s anti-derivative is,
= [Tex]\int f(x)dx[/Tex]
= [Tex]\int 2e^xdx[/Tex]
Using the property 1 mentioned above,
= [Tex]2\int e^xdx[/Tex]
= 2ex + C
Example 3: Find the integral for the given function f(x), f(x) = 5x-2
Solution:
Given f(x) = 5x-2
Using reverse power rule
= [Tex]\int f(x)dx[/Tex]
= [Tex]\int 5x^{-2}dx[/Tex]
Using property 1 mentioned above,
= [Tex]5\int x^{-2}dx[/Tex]
= [Tex]\frac{-5}{x} + C[/Tex]
Example 4: Find the integral for the given function f(x), f(x) = sin(x) + 5cos(x)
Solution:
Given f(x) = sin(x) + 5cos(x)
sin(x) and cos(x) are standard functions, and its integral is,
= [Tex]\int f(x)dx[/Tex]
= ∫ (sin(x) + 5cos(x))dx
= [Tex]\int sin(x)dx + 5\int cos(x)dx[/Tex]
= [Tex]-cos(x) + 5sin(x) + C[/Tex]
Example 5: Find the integral for the given function f(x), f(x) = 5x-2 + x4 + x
Solution:
Given f(x) = 5x-2 + x4 + x
Using reverse power rule
= [Tex]\int f(x)dx[/Tex]
= [Tex]\int (5x{-2} + x^4 + x)dx[/Tex]
= [Tex]\int (5x{-2} + x^4 + x)dx[/Tex]
= [Tex]5\int x^{-2}dx + \int x^4dx + \int xdx[/Tex]
= [Tex]\frac{-5}{x} + \frac{x^5}{5} + \frac{x^2}{2}[/Tex]
Example 6: Is the graph given below differentiable or not?
Solution:
Graph given above is, y = 4 is a constant graph.
And constant graph are easily differentiable.
Integration Formulas
Indefinite Integrals: The derivatives have been really useful in almost every aspect of life. They allow for finding the rate of change of a function. Sometimes there are situations where the derivative of a function is available, and the goal is to calculate the actual function whose derivative is given.
In this article, we will discuss Indefinite Integrals, graphical interpretation, formulas, and properties.
Table of Content
- What are Indefinite Integrals?
- Graphical Interpretation of Integrals
- Integrals by Graphs
- Calculating Indefinite Integral
- All Formulas of Indefinite Integrals
- Properties of Indefinite Integrals
- Property of Sum
- Property of Difference
- Property of Constant Multiple
- Difference Between Indefinite Integral and Definite Integral
- Indefinite Integrals Examples
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