Indefinite Integration

Integrals have numerous uses, especially in engineering subjects involving maths and physics. Indefinite integration is solvable using various types of integrations. It uses parts, substitution, partial fractions, and trigonometric functions to find the area. To get detailed knowledge about indefinite integration, we should know what indefinite integration is and its types.

Integration

Integration is the process of determining the integral function using various integral notations. It is the reverse process or the opposite of differentiation. It is often mentioned as the antiderivative.

Integration can be classified as:

(i) Definite integrals

(ii) Indefinite integrals

If an integral has lower and upper limits, it is called a definite integral. An indefinite integral has no limits. In this case, we have to determine the limits of the integral on our own using various formulas.

Indefinite Integration

Indefinite integration is the process of inverting the differentiation. It usually has no upper and lower limits. It is famously known as the antiderivative function. 

In mathematical terms, we use the following expressions:

Integral of f with respect to x — ∫f(x) dx

The antiderivative of f(x) —- F(x)

So the expression will be:

∫f(x) dx = F(x) + constant

The constant can be obtained in a discretionary way from the real numbers, which helps obtain certain functions that form the antiderivatives’ function. In other words, even though it can be calculated in many different ways, the constant helps obtain different integrals to the provided function. 

For example, consider the equation, f(x) = 5x4. We have to find F(x). 

We can consider that F'(x) = 5x4, and by using the power rule, we can reverse the process to obtain the solution.

F(x) = x5. And the constant will be 0 in case we differentiate it. So, apply the same in the given equation.

F(x) = x5 – 16

F(x) = x5 + 1234567

F(x) = x5 + π

So, if we differentiate the above equation, the answer will be

f(x) = 5x3

Instead of getting a set with boundary values, we can find an equation that would produce the integral due to differentiation without using the values to get a definite answer.

Consider the equation f(x) = 3x2. We have to find an equation F(x) so that F’(x) = 3x2. We can use the power rule from differentiation to reverse and obtain F(x) = x3. However, this is not the only answer. If we differentiate a constant value, then we can get the result as zero (0). Thus, the function can be written as follows:

F(x) = x3 – 16

F(x) = x3 + 1234567

F(x) = x3 + p

So, there are many ways to obtain the solution. The major difference is by adding the constant into the expression, as seen in the above example.

Examples of Indefinite Integrals 

The following examples explain how an indefinite integral works, making the process easy to find the solution.

Example 1:

Determine the provided indefinite integral: ∫2x5 -6x2 +9 dx

Solution:

Given function,

∫ 2 x5 – 6 x2 + 9 dx

By integrating the provided function, we arrive 

∫ 2 x5 – 6 x2 + 9 dx = 2 (x6/2) – 6(x3/3) + 9x + C

We have to add constant whenever we integrate the function,

By doing simplification, we can easily get the following function

Hence, ∫2 x5 – 3 x2 + 9 dx = x6 – 3 x3 + 9x + C

Example 2: 

Determine f(x), provided that f ‘(x) = 8x8 -40x4 + x2 + 5

Solution:

Given function,

f ‘(x) = 8x8 -40x4 + x2 + 5

Applying the inverse process to get the integration,

Hence, f(x) = ∫f ‘(x) dx=∫[8x8 -40x4 + x2 + 5] dx

f(x) = (2/4) x9 – 8x5 +(1/3) x3 + 5x+ C

Applications of Integral Calculus

Integral calculus is vital in many fields of both mathematics and the sciences. The following are some major uses of integral calculus. Integration was essential in the discovery of:

• The space that exists between two curves

• The centre of gravity

• Kinetic energy and its calculations

• Work in physics

• Distance, speed, and acceleration in physics

The integral approach is used to sum functions on a large scale. Finding the area between curves, volume, the average value of the function, kinetic energy, centre of mass, work-done, and other uses of integrals are numerous in the real world.

Conclusion

Determining the antiderivative of a function is important in calculus. Assume that there exists an integral that is found between boundaries. Then, reversing the differentiation process, we can arrive at the indefinite integral.