Introduction to integral, its rules and how we find it:

In mathematics, an integral is a mathematical article that can be taken as simplifying the area under the curve. It is a way of expressing a function as a sum of infinitely many terms, each of which is a product of the function and a small portion of the domain of the function.

Here I introduce it very simple way like if your friend gives you a wooden stick. He asks you to break it. Can you do so? Yes, it will be very easy for you to do so because it’s a single stick. But when he gives you nine to ten sticks to break?

No, it can’t be easy to break it. Because the number of sticks increases it is difficult to break them. The way of unity thing is an integration of things. Similarly, in mathematics, we have an integration of two or three functions.

Moreover, in this article, the basic definition of integral and its rules and with the help of formulas integration problems will be solved.

Definition of integral

“Integral is defined as the limit of a sum of areas of infinitesimally small rectangles or as the limit of a sum of volumes of infinitesimally small cubes”

Formula of integral

∫f (t) d(t) = F(t) + C

In general, the integral of a real-valued function f (t) concerning a real variable x on an interval [x, y] is written as.

Rules of integral:

There are a few main rules that are necessary to be followed while integrating a function to find its integral. These rules are discussed below:

Difference Rule.
Sum Rule.
Constant rule.
Multiplication by Constant.
Product Rule
Power Rule.

1. Difference Rule:

The integral of a difference is equal to the difference of their integral. For better understanding, it can be expressed as

∫ [f (t) – g (t)] d (t) = ∫ f (t) d (t) – ∫ g (t) d (t)

2. Sum Rule:

The Integral of a sum is equal to the sum of its integral. For better understanding, it can be expressed as

∫ [f (t) + g (t)] d (t) = √ f (t) d (t) + ∫ g (t) d (t)

3. Constant Rule:

It is a common rule used in integration and also in derivation that a scalar can be taken out of an integral in this rule let’s suppose p is a scalar then

∫ {f (t) * q} d (t) = q ∫ f (t) d (t)

4. Product Rule:

It is a common rule used in integration and also in derivation when we integrate two functions at a time then we use this function

∫ x sin(x)dx =x ∫ sin(x)dx+∫ d/ dx (x)∫ sin(x)dx

5. Power Rule:

It is also a common rule which is used in integration and it helps to solve integration expressions with radicals in them.

∫ xⁿ d (t) = x^{n + 1}/ (n+1) +c

Integral of Trigonometric functions

There are six basic trigonometric functions with which you are familiar sine used as sin, cosine and used as cos, tangent used as tan, cosecant used as cosec, secant used as a sec, and cotangent used as cot where cosec, sec, and cot are reciprocal of sin cos and tan.

The integral of cos(x) is equal to sin(x) + c where c is an arbitrary constant.

∫cos (x) d (x) = sin (x) + c

The integral of sin (x) is equal to – cos(x) + c where c is an arbitrary constant.

∫sin (x) d (x) = -cos (x) + c

The integral of tan (x) is equal to – ln |cos(x)| + c where c is an arbitrary constant.

∫tan (x) d (x) = – ln |cos(x)| + c

The integral of sec (x) is equal to ln |sec(x) +tan(x)| + c where c is an arbitrary constant.

∫sec (x) +tan (x) d (x) = ln |sec(x) +tan(x)| + c

The integral of cosec (x) is equal to – ln |cosec(x) +cot(x)| + c where c is an arbitrary constant.

∫ cosec (x) d (x) = – ln |cosec(x) +cot(x)| + c

The integral of cot (x) is equal to ln |sin(x)| + c where c is an arbitrary constant.

∫ cot (x) d (x) = ln |sin(x)| + c

Similarly, inverse trigonometric functions

∫ 1 / √ (1 – x²) d (x) = sin ^-1(x) + C, where x ≠ ±1 and c is an arbitrary constant.
∫ -1 / √ (1 – x²) d (x) = cos ^-1(x) + C, where x ≠ ±1and c is an arbitrary constant.
∫ 1 / (1 + x ²) d (x) = tan ^-1(x) + C, where c is an arbitrary constant.
∫ -1 / (1 + x²) d (x) = cot ^-1(x) + C, where c is an arbitrary constant.
∫ -1 / |x| (1 – x²) d (x) = cosec ^-1(x) + C, where x ≠ ±1, 0 and c is an arbitrary constant.
∫ 1 / |1| √ (1 – x²) d (x) = sec ^-1(x) + C, where x ≠ ±1, 0 and c is an arbitrary constant.

Example Section:

In this section, with the help of examples, the topic is explained.

Example 1:

Integrate the given function f (x) = x+1

Solution:

Step 1:

With the help of a formula

F(x) = ∫ f (x) dx + C, where c is an arbitrary constant.

Step 2:

F(x) =∫ (x+1) dx

Now, with help of the formula

F(x) = ∫ x dx+∫1dx

=x²/ 2 + x + c

The above problem can also be integrated with the help of an integral calculator by Meracalculator to get the result without involving into the lengthy and tricky calculations.

Example 2:

Find the integral of 2 sin(x).

Solution:

Step 1:

f (x) =2 sin(x)

with the help of a formula