The document discusses indefinite integrals and their relationship to derivatives. It states that the indefinite integral is the inverse of the derivative, so finding the integral of a function means finding its antiderivative. The document provides examples of using formulas to find the antiderivatives of functions like 5x^4 by recognizing it as the derivative of x^5. It also demonstrates how integrals follow properties like scalar multiplication and addition/subtraction.

1. INTEGRAL
Indefinite Integral
In calculus integral known are two concept,namely indefinite
integral and definite integral.Concept of the indefinite integral
is the inverse of derivative.
Integraling a derivative function f(x)=f’(x) means we seek
integral or anti-derivtive,is f(x),if the function is derived
produce original function f(x).
If F’(x) = f(x) so F(x) is an anti-derivative from f(x)
Example :find the anti-derivative from =5x4

2. Solution
Determine the function f(x) derivatives is f(x) = 5x4.we know
that if F(x) = xn,so F’(x) =f(x) =nxn-1.so that if
nxn-1=5x4
so n-1= 4 and n = 5
so that,x5 is an anti-derivatiive of 5x4
general integral of function f written

3. Example : Compute the integral of the following
functions
3dx
Solution : we will just plug into the formula,and the
integral above n= 3.
3 dx = x3+1 + C
= x4 + C

4. Integral scalar multiplication
for every real number k
Example :compute the integral of the following function
3 dx
Solution :we will just plug into the formula
3 dx =2 3 dx
=2 ( x4 ) + C
= x4 + C

5. Integral use Addition and Reduction
Example :find the integral of the following finction
2
- 4x +5) dx
Solution : we will just plug into the formula
2 – 4x + 5) dx = 3 2 dx – 4 +5
= 3 ( x3 ) – 4( x2) + 5x + C
= x3 – 2x2 + 5x + C