Integrals of Some Particular Functions

Introduction

Integration, in English, refers to the act of bringing disparate parts together to form a whole. In integral calculus in mathematics, a known differential is used to find a function. Integration is the opposite process of differentiation. 

The area of a region encompassed by a given graph of functions is defined and calculated using integration. 

Integration was once known as the exhaustion method, was eventually renamed integration. 

There are two types of integrals in integral calculus: Indefinite integrals and definite integrals. 

Differentiation and integration are foundational tools in mathematics used to solve many different real-life problems in the sciences. 

The scientist Leibnitz was the first to derive and propose the method of integration. 

Integration

Integration is a formulaic method of determining the area of the region under a given curve function. The sum of an integration function approaches a limit equal to the function’s region under the curve. 

Practically, such an area would be determined by drawing as many little rectangles as possible to cover the area and then adding their respective areas together. The integration enables this calculation in a much faster and easier way.

The process of obtaining a function anti-derivative is known as integration. If a function is integrable, it is known as a definite integration. This means that its integral over the domain is finite and that the bounds are clearly defined. 

Integration as Inverse Differentiation 

If a function’s derivative is given and its primitive or original function needs to be derived, the process of integration will be used. Anti-differentiation or integration is the term for this type of process. 

When given a function’s derivative, determining the original function is known as integration. Integrals and derivatives are opposites of each other. 

Take the function f(x)=sin x. 

The derivative of f(x) will be expressed as: f'(x) = cos x 

The function cos x is known as the derived function of sin x. 

Integration Techniques

While integrals can sometimes be derived through visual inspection, it is mostly found to be insufficient in determining the integral of a function. 

There are other ways to obtain the integral of a function that is reduced to its standard form. These methods include:

• Method of decomposition
• Substitutional integration
• Integration using partial fractions
• Parts-based integration

Integral Calculus

The values of the function obtained through the integration process are known as integrals. In other words, integration is the process of obtaining f(x) from F(x).

Integrals allocate numbers to functions in a way that represents values of displacement and motion, values of area and volume, and other values that arise from combining small amounts of data. 

The function f can be found by using the derivative f’ of the function. Integral calculators online automate this process as well. 

The function f is referred to as an anti-derivative or integral of f’ in this case.

Example: if f(x) = x2, its derivative will be expressed as:

f'(x) = 2x = g(x),

when g(x) = 2x, then anti-derivative of g(x) = ∫ g(x) = x2

Definite Integral

A definite integral is an integral that contains both upper and lower boundaries. A definite integral may be expressed mathematically as follows:

abf(x)dx

Indefinite Integral

Indefinite integrals are those that do not have upper or lower bounds. They may be expressed mathematically as follows:

∫f(x)dx = g(x) + C

The function f(x) is referred to as the integrand, and C is a constant.

Characteristics of Indefinite Integrals

The integrand is the derivative of an integral. ∫ f(x) dx = g(x) +C.

Given that two indefinite integrals with the same derivative produce the same family of curves, they are considered comparable.

∫ [ f(x) dx – g(x) dx] =0

The sum or difference of the integrals of a finite number of functions equals the sum or difference of the integrals of the individual functions. This may be mathematically expressed as: 

∫ [ f(x) dx +g(x) dx] = ∫ f(x) dx + ∫ g(x) dx

The constant multiplied with the function is taken out of the integral.∫ k f(x) dx = k ∫ f(x) dx, where k ∈ R.

The form is arrived at by combining the previous two attributes:

 ∫ [k1f1(x) + k2f2(x) +… knfn(x)] dx = k1∫ f1(x)dx + k2∫ f2(x)dx+ … kn ∫ fn(x)dx

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.

Integrals of Some Special Functions

Here are the integrations of some particular functions expressed mathematically: 

∫1×2 – a2 dx = 12alog(x-ax+a)+ C

∫1a2 – x2 dx = 12alog(a+xa-x) + C

∫1×2 + a2dx = (1a) tan-1(xa) + C

∫1×2 – a2 dx = log|x + x2 – a2 | + C

∫1a2 – x2) dx = sin-1(xa) + C

∫1×2 + a2  dx = log| x + √x2 + a2 | + C

Conclusion

The effectiveness of numerical integration is influenced by a variety of practical considerations. It is especially useful as simple model functions may not always accurately mimic the behaviour of unknown functions.

It is, of course, challenging to work with complicated model functions. While there are some circumstances wherein a straightforward method for determining how accurate a numerical integral may be used, this can be rather complicated in general.

Nonetheless, scientists, mathematicians, and engineers all use integral functions and numerical integration by applying them in various contexts.