The document discusses various types of integrals and rules for finding antiderivatives. It defines definite and indefinite integrals. It then lists and explains the main antiderivative rules for powers, chain rule, product rule, quotient rule, scalar multiples, sums and differences, trigonometric functions, and inverse trigonometric functions. Examples are provided to illustrate each rule.

1. Mathematics
Project By:-
Yuvraj Singh
Aryansh Agarwal
Ajay Agarwal
Rahul Choudhary
X I I Science

2. Integration
Integration is the calculation of an integral. Integrals in maths are used
to find many useful quantities such as areas, volumes, displacement, etc.
When we speak about integrals, it is related to usually definite integrals.
The indefinite integrals are used for antiderivatives. The integration
denotes the summation of discrete data. The integral is calculated to find
the functions which will describe the area, displacement, volume, that
occurs due to a collection of small data, which cannot be measured
singularly.

3. Definite Integral
An integral that contains the upper and
lower limits then it is a definite
integral. On a real line, x is restricted
to lie. Riemann Integral is the other
name of the Definite Integral.
A definite Integral is represented as:
Indefinite Integral
Indefinite integrals are defined
without upper and lower limits. It is
represented as:
∫f(x)dx = F(x) + C
Where C is any constant and the
function f(x) is called the integrand.

4. Integrals(Anti-Derivatives )

5. List of Anti-Derivative Rule
The list of most commonly used antiderivative rules for the product,
quotient, sum, difference, and the composition of functions is as
follows:
● Antiderivative Power Rule
● Antiderivative Chain Rule
● Antiderivative Product Rule
● Antiderivative Quotient Rule
● Antiderivative Rule for Scalar Multiple of Function
● Antiderivative Rule for Sum and Difference of Functions

6. 1. Anti-Derivative Power Rule
The antiderivative rule of power of x is given by ∫xn dx = xn+1/(n +
1) + C, where n ≠ -1. This rule is commonly known as the
antiderivative power rule.
● ∫x2 dx = x2+1/(2+1) + C = x3/3 + C
● ∫x-4 dx = x-4+1/(-4+1) + C = x-3/(-3) + C = -x-3/3 + C

7. 2. Anti-Derivative Chain Rule
The chain rule of derivatives gives us the antiderivative chain rule which is
also known as the u-substitution method of antidifferentiation. The
antiderivative chain rule is used if the integral is of the form ∫u'(x) f(u(x))
dx.
Solve ∫2x cos (x2) dx
Solution: Assume x2 = u ⇒ 2x dx = du. Substitute this into the integral,we
have
∫2x cos (x2) dx = ∫cos u du
= sin u + C
= sin (x2) + C

8. 3. Anti-Derivative Product Rule
It is one of the important antiderivativerulesand is used when the antidifferentiation of the product of
functionsis to be determined. The formula for theantiderivativeproduct rule is ∫f(x).g(x) dx = f(x) ∫g(x) dx
− ∫(f′(x) [ ∫g(x) dx)]dx +C.
This method is also commonly known as the ILATE or LIATE method of integration which is abbreviated of:
● I - Inverse Trigonometric Function
● L - LogarithmicFunction
● A - AlgebraicFunction
● T - Trigonometric Function
● E - Exponential Function
∫x ln x dx = ln x ∫x dx - ∫[(ln x)' ∫x dx] dx
= (x2/2) ln x - ∫(1/x)(x2/2) dx
= (x2/2) ln x - ∫(x/2) dx
= (x2/2) ln x - x2/4 + C

9. 4. Anti-Derivative Quotient Rule
The antiderivativequotient rule is used when the function is given in the form of numerator
and denominator. If the function includes algebraic functions, then we can use the
integration by partial fractions method of antidifferentiation.
consider a function of the form f(x)/g(x). Now, differentiating this we have,
d(f(x)/g(x))/dx = [f'(x)g(x) - g'(x)f(x)]/[g(x)]2
Now, integrating both sides of the above equation, we have
f(x)/g(x) = ∫{[f'(x)g(x) - g'(x)f(x)]/[g(x)]2} dx
= ∫[f'(x)/g(x)] dx - ∫[f(x)g'(x)/[g(x)]2] dx
⇒ ∫[f'(x)/g(x)] dx = f(x)/g(x) + ∫[f(x)g'(x)/[g(x)]2] dx
If f(x) = u and g(x) = v, then we have the antiderivative quotient rule as:
∫du/v = u/v + ∫[u/v2] dv

10. 5. Anti-Derivative for Scalar Multiple
To find the antiderivative of scalar multiple of a function f(x), we can
find it using the formula given by, ∫kf(x) dx = k ∫f(x) dx. This implies,
the antidifferentiation of kf(x) is equal to k times the
antidifferentiation of f(x), where k is a scalar. An example using this
antiderivative rule is:
∫4x dx = 4 ∫xdx
= 4 × x2/2 + C
= 2x2 + C

11. 6. Anti-Derivative for Sum and Difference
When the antidifferentiation of the sum and difference of functions is to
be determined, then we can do it by using the following formulas:
● ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
● ∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx
Some of the examples of the antiderivative rule for sum and difference
of functions are as follows:
● ∫[4 + x2] dx = ∫4 dx + ∫x2 dx = 4x + x3/3 + C
● ∫(sin x - log x) dx = ∫sin x dx - ∫ log x dx = -cos x - x log x +
x + C

12. Anti-Derivative Rule for Trigonometric Functions
We have six main trigonometric functions, namely sine, cosine, tangent, cotangent,
secant, and cosecant. Now, we will explore their antiderivative rules of these
trigonometric functions as follows:
● ∫sin x dx = -cos x + C
● ∫cos x dx = sin x + C
● ∫tan x dx = ln |sec x| + C
● ∫cot x dx = ln |sin x| + C
● ∫sec x dx = ln |sec x + tan x| + C
● ∫csc x dx = ln |cosec x - cot x| + C

13. Anti-Derivative Rule for Inverse Trigonometric
Functions
We have six main inverse trigonometric functions, namely inverse sine, inverse cosine,
inverse tangent, inverse cotangent, inverse secant, and inverse cosecant.Now, we will
explore their antiderivative rules of these trigonometric functions as follows:
● ∫sin-1x dx = x sin-1x + √(1 - x2) + C
● ∫cos-1x dx = x cos-1x - √(1 - x2) + C
● ∫tan-1x dx = x tan-1x - (1/2) ln(1 + x2) + C
● ∫cot-1x dx = x cot-1x + (1/2) ln(1 + x2) + C
● ∫sec-1x dx = x sec-1x - ln |x + √(x2 - 1)| + C
● ∫csc-1x dx = x csc-1x + ln |x + √(x2 - 1)| + C