Proofs of Integration Formulas

The integration of a function f(x) is given by F(x) and it is given as:

∫f(x)dx = F(x) + C

Here R.H.S. of the equation means integral of f(x) with respect to x.

F(x)is called anti-derivative or primitive.

f(x)is called the integrand.

dx is called the integrating agent.

C is an arbitrary constant called as the constant of integration.

x is the variable of integration.

The anti-derivatives of basic functions are known to us. The integrals of these functions can be obtained readily. In the upcoming discussion let us discuss few important formulae and their applications in determining the integral value of other functions.

1.Integration

Proof: The integrand can be expressed as:

Integration Formulas

Multiplying the numerator and the denominator by 2a and simplifying the obtained expression we have;

Integration Formulas

Therefore, upon integrating the obtained expression with respect to x, we have;

Integration Formulas

According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,

Integration Formulas

Therefore equation 1 can be rewritten as:

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Proof: The integrand can be expressed as:

Integration Formulas

Multiplying the numerator and the denominator by 2a and simplifying the obtained expression we have;

Integration Formulas

Therefore, upon integrating the obtained expression with respect to x, we have;

Integration Formulas

According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,

Integration Formulas

Therefore equation 2 can be rewritten as:

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Proof: Let x = a tan Ɵ. Differentiating both sides of this equation with respect to x we have;

dx = a sec2Ɵ dƟ

Therefore, using this, the integral can be expressed as:

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Proof: Let x = a sec Ɵ. Differentiating both sides of this equation with respect to x we have;

dx = a secƟ tan Ɵ dƟ

Therefore, using this, the integral can be expressed as:

Integration Formulas

Using the trigonometric identity sec2Ɵ– 1 = tan2Ɵ, the above equation can be written as

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Substituting the value of Ɵ in the above equation we have;

I 20

Here, C = C1 – log |a|

5.Integration Formulas

Proof: Let x = a tan Ɵ. Differentiating both sides of this equation with respect to x we have;

dx = a sec2 Ɵ dƟ

Therefore, using this, the integral can be expressed as:

Integration Formulas

Using the trigonometric identity sec2Ɵ = 1 + tan2Ɵ, the above equation can be written as

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Substituting the value of Ɵ in the above equation we have;

Integration Formulas

Here, C = C1 – log |a|

6.   Integration

Proof: Let x = a sin Ɵ. Differentiating both sides of this equation with respect to x we have;

dx = a cosƟ dƟ

Therefore, using this, the integral can be expressed as:

Integration Formulas

Using the trigonometric identity 1 – sin2Ɵ =cos2Ɵ, the above equation can be written as

Integration Formulas

Integrating with respect to x, we have

Integration Formulas

Substituting the value of Ɵ in the above equation we have;

Integration Formulas

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Practise This Question

In the below figure,

___ angles are present and they are ___