Integration in Maths

The process of determining the function from its derivative is called Integration. In other words, the procedure of finding the anti-derivatives of the function is called the integration. The result obtained after the integration is called integral. The integration can be done using multiple methods like integration by substitution, integration by parts, integration by partial fraction, etc.

If f is the positive continuous function defined over an interval [a, b] then, the area between the function f graph and x-axis results in the integration of f w.r.t x. The area under the curve gives the definite integration of f.

Integration Symbol

The symbol of integration is ∫. For the definite integral we apply limits and use symbol ∫^a_b .

Antiderivative: Integration as Inverse Process of Differentiation

The process of finding the antiderivative i.e., the inverse of the derivative is called integration. If Φ(x) is a function and the derivative of Φ(x) is f(x) then, integration of f(x) results in Φ(x).

(d / dx) {Φ(x)} = f(x) ⇔ ∫f(x) dx = Φ(x) + C

OR

∫[d/dx]g(x) dx = g(x)

Rules for Integration

Some important rules of integration are:

• Power Rule

The power rule of integration is stated as:

∫xⁿ dx = xⁿ⁺¹/ (n + 1)

• Addition Rule

The addition rule of integration is stated as:

∫{f(x) + g(x)} dx = ∫f(x) dx + ∫g(x) dx

• Subtraction Rule

The subtraction rule of integration is stated as:

∫{f(x) - g(x)} dx = ∫f(x) dx - ∫g(x) dx

• Constant Multiple Rule

The multiplication of constant rule of integration can be stated as:

∫k f(x) dx = k ∫f(x) dx,

Where k is constant.

Integration Formulas

The various integration formulas are:

1. \frac{d}{dx}{Φ(x)} = f(x) ⇔ ∫f(x) dx = Φ(x) + C
2. ∫xⁿ dx = xⁿ⁺¹/ (n + 1) + C
3. ∫(1 / x) dx = log_e|x| + C
4. ∫e^x dx = e^x + C
5. ∫a^x dx = [a^x/ log_ea] + C
6. ∫sin x dx = -cos x + C
7. ∫cos x dx = sin x + C
8. ∫sec²x dx = tan x + C
9. ∫cosec²x dx = -cot x + C
10. ∫sec x tan x dx = sec x + C
11. ∫cosec x cot x dx = - cosec x + C
12. ∫tan x dx = ln |sec x| + C = -ln |cos x| + C
13. ∫cot x dx = ln |sin x| + C
14. ∫sec x dx = ln |sec x + tan x| + C
15. ∫cosec x dx = ln |cosec x - cot x| + C
16. ∫\frac{1}{\sqrt{a^2 -x^2}}dx = sin^-1(x/a) +C
17. ∫- \frac{1}{\sqrt{a^2 -x^2}}dx = cos^-1(x/a) + C
18. ∫ \frac{1}{{a^2 + x^2}}dx = (1/a) tan^-1 (x/a) + C
19. ∫- \frac{1}{{a^2 + x^2}}dx = (1/a) cot^-1(x/a) + C
20. ∫\frac{1}{x\sqrt{x^2 -a^2}}dx = (1/a) sec^-1(x/a) + C
21. ∫- \frac{1}{x\sqrt{x^2 -a^2}}dx = (1/a)cosec^-1(x/a) + C

Learn more about Integration Formulas.

Integration Techniques

There are various methods to find the integration of a function. Some of these are listed below:

• Integration by Substitution
• Integration by Parts
• Integration by Partial Fraction

Read more about Method of Integration.

Integration of Basic Functions

There are different integration formulas for different functions. Below we will discuss the integration of different functions in depth and get complete knowledge about the integration formulas.

Integration of Constant Function

The integration of a constant function is given by:

∫k dx = kx + C, where k is constant

Integration of Trigonometric Functions

The integration of trigonometric functions is given by:

• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C
• ∫cosec²x dx = -cot x + C
• ∫sec x tan x dx = sec x + C
• ∫cosec x cot x dx = - cosec x + C
• ∫tan x dx = -log |cos x| + C
• ∫cot x dx = log |sin x| + C
• ∫sec x dx = log |sec x + tan x| + C
• ∫cosec x dx = log |cosec x - cot x| + C

Integration of Exponential and Logarithmic Functions

The integration of exponential and logarithmic function is given by:

• ∫(1 / x) dx = log_e|x| + C
• ∫e^x dx = e^x + C
• ∫a^x dx = [a^x/ log_ea] + C

Integration vs Differentiation

The basic difference between integration and differentiation is tabulated below:

Applications of Integration

There are various applications of integration. Some of them are listed below:

• Integration is used to find area under the curve.
• It is also used to find the volumes.
• Integration is used to find area between the two curves.
• It is used in multiple formulas in physics.

Integration in Physics and Engineering

Integration is used widely in Physics and Engineering

• In Physics integration is used in multiple formulas for example finding the velocity from acceleration and many more.
• In Engineering integration is used in different fields and in many formulas of engineering mechanics etc.

Integration in Economics and Finance

Integration is also helpful in Economics and Finance to calculate marginal and total revenue, costs, profits, consumer and producer surplus, capital accumulation over a specified time and in the Lorenz curve and Gini coefficient.

Related Articles:

• Calculus in Maths
• Integral Calculus
• Differentiation and Integration Formula
• Integration Formulas
• Methods of Integration
• Application of Integration

Solved Examples on Integration

Example 1: Solve: ∫x⁶dx

Solution:

∫xⁿ dx = xⁿ⁺¹/ (n + 1) + C

∫x⁶ dx = x⁶⁺¹/ (6 + 1) + C

∫x⁶ dx = [x⁷/ 7] + C

Example 2: Solve: ∫3^x+2dx

Solution:

∫3^x+2dx = ∫3^x+2dx

∫3^x+2dx = ∫3^x.3²dx

∫3^x+2dx = 9 ∫3^xdx

∫3^x+2dx = 9[3^x / log_e3] + C

Example 3: Solve: ∫(x³ + 4x² + 3x + 1)dx

Solution:

∫(x³ + 4x² + 3x + 1)dx = ∫x³ dx + ∫(4x²) dx + ∫(3x) dx + ∫(1) dx

∫(x³ + 4x² + 3x + 1)dx = [x³⁺¹ / (3 + 1)] + 4 [x²⁺¹ / (2 + 1)] + 3[x¹⁺¹ / (1 + 1)] + x + C

∫(x³ + 4x² + 3x + 1)dx = [x⁴ / 4] + 4 [x³ / 3] + 3[x² / 2] + x + C

Example 4: Solve: ∫(x + 3sinx)dx

Solution:

∫(x + 3sinx) dx = ∫x dx + ∫ (3sinx) dx

∫(x + 3sinx) dx = [x¹⁺¹ / (1 +1)] + 3(-cos x) + C

∫(x + 3sinx) dx = (x² / 2) - 3 cos x + C

Example 5: Solve: ∫[1/ (x² + 25)] dx

Solution:

∫[1/ (x² + 25)] dx = ∫[1/ (x² + 5²)] dx

∫[1/ (x² + 25)] dx = (1 / 5) tan^-1 (x / 5) + C

Example 6: Solve: ∫(2e^x + x³) dx

Solution:

∫(2e^x + x³) dx = ∫2e^x dx + ∫x³ dx

∫(2e^x + x³) dx = 2e^x + [x³⁺¹ /(3 +1)] + C

∫(2e^x + x³) dx = 2e^x + [x⁴ /4] + C

Practice Questions on Integration

Question 1: Evaluate: ∫ (x⁴ + e^x + 3sinx) dx

Question 2: Calculate: ∫sin x. cos x dx

Question 3: Simplify: ∫1 / {8 √(x² - 64)} dx

Question 4: Calculate: ∫(sin x - cos x) / (sin x + cos x) dx

Question 5: Evaluate: ∫ tan 5x dx

Question 6: Simplify: ∫ [1 / (x² + 2)] dx

Question 7: Find: ∫ x e^x dx

Question 8: Evaluate: ∫ e^2x dx

Question 9: Calculate: ∫ (1 + x²)^1/2 dx

Question 10: Simplify: ∫ (1 + cos 2x) dx