Integration Rules: Basic Rules, Formulas & Solved Examples

Integration Rules are the rules used to integrate different types of functions. Integration is a means to calculate many useful quantities such as area, volume central point, and other kinds of parameters. The area beneath the curve in a graph is the most popular application of integration.

• Integration rules are used to solve complicated functions in integration.
• In addition to the integration rules, there are integration formulas for the purpose of substituting the integral form.
• Integration occurs when a derivative is already provided and the function needs to be derived through such a derivative accordingly.
• It is also known as anti-differentiation, as it is the inverse of differentiation. 

Read More: NCERT Solutions For Class 12 Mathematics Chapter 7 Integrals

Key Terms: Integration, Functions, Integration Rules, Derivative, Differentiation, Power Rule, Integration by Parts, Trigonometry

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What are Integration Rules?

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Integration Rules are applied for integrating various kinds of functions. They are used to find the integral easily. 

• For example, ∫ 2y dy = y² + C as d/dy (y²) = 2y.
• Such an outcome is derived through the Power Rule of Integration where ∫yn dy = (yⁿ⁺¹/n+1) + C.
• Here, C the integration constant is C, which is added after the integral of any kind of integration function. 

Integral and Derivative

The Integration Rules are mentioned as follows:

Power Rule of Integration

In accordance with the power rule of integration, if y is raised to the power n is integrated, the result is 

Example: Integrate ∫y⁴ dy.

 ∫y⁴ dx = y⁽⁴⁺¹⁾/(4+1) = x⁵/5

Sum Rule of Integration

In accordance with the sum rule of integration, integrating the sum of two functions is equal to the sum of the integral of each function.

Example: ∫ (y + y³ )dy 

= ∫y dy + ∫y³ dy

= y²/2 + y⁴/4 + C

Integration by Special Functions Video Explanation

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Difference Rule of Integration

Difference rule of integration bears similarity to the sum rule of integration. In accordance with the sum rule of integration, integrating the subtracted outcome of two functions equals the subtracted outcome of the integrals of the functions.

Example: ∫ (y - y³ )dy 

= ∫y dy - ∫y³ dy

= y²/2 - y⁴/4 + C

Multiplication by Constant 

When a constant and a function are multiplied with each other, the result is

Example: ∫4 y.dy

= 4 ∫ y.dy

= 4 y²/2 + C

= 2 y² + C

Product Rule 

The product rule is another term for the method “Integration by Parts”. It is applied when the multiplication of two functions takes place. 

Here, 

• The function of u(y) is u.
• The function of v(y) is v.
• The derivative of the function u(y) is u'.

Read More: Continuous Integration

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Integration Rules of Basic Functions

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Integration rules are applicable to different types of functions. Given below are the basic rules for integration of the some common functions, such as:

• Constant
• Variable
• Square
• Reciprocal
• Exponential
• Trigonometry

Integration of Constant

The result of integrating the constant function would be 

Example: ∫4 dx = 4x + C

Integration of Variable

If the variable provided is y, it can be denoted as 

Integration of Square

If the function provided is a square, then the expression would be 

Integration of Reciprocal 

When 1/y is a reciprocal function of y, then the expression would be 

Integration of Exponential Function

Integration rules of various exponential functions are as follows: 

• ∫ f^x dx = f^x + C
• ∫ b^x dx = b^x / ln (b) + C
• ∫ ln (y) dy = y ln(y) − x + C

Integration of Trigonometric Function

The rules of Trigonometry are as follows: 

• ∫ cos (y) dy = sin (y) + C
• ∫ sin (y) dx = - cos (y) + C
• ∫ sec²(y) dx = tan (y) + C

Read More: Integral Formulas

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Integration Rules of Trigonometric Functions

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The six Trigonometric Functions include sin, cos, tan, cosec, sec and cot. The Integration Rules of Trigonometric Functions are mentioned as follows: 

• Integration of sin y - ∫ sin y dy = - cos y + C
• Integration of cos y - ∫ cos y dy = sin y + C
• Integration of tan y - ∫ tan y = In (sec y) + C (or) - In I(cos y) + C
• Integration of cosec y - ∫ cosec y dy = In I cosec y - cot y I + C (or) - In I cosec y + cot y I + C (or) In I tan (x/2) I +C
• Integration of sec y - ∫ sec y dy = In I sec y + tan y I + C (or) (½) In I (1 + sin y) / (1 - sin y) (or) In I tan [x/2 (π/4) ] | + C
• Integration of cot y - ∫ cot y dy = In I sin y I + C

In addition to the above, other Integration Rules of Trigonometric Functions are mentioned as follows: 

• ∫ sec²y dy = tan y + C
• ∫ cosec²y dy = - cot y + C
• ∫ sec y. tan y dy = sec y + C
• ∫ cosec y . cot y dy = - cosec y + C

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Integration Rules of Inverse Trigonometric Functions

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The six Inverse Trigonometric Functions include arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹⁾, arccosec (cosec⁻¹), arcsec (sec⁻¹) and arccot (cot⁻¹). The Integration Rules of Inverse Trigonometric Functions are mentioned as follows: 

• ∫ sin^-1y dy = y sin^-1y + √(1 - y²) + C
• ∫ cos^-1y dy = y cos^-1y - √(1 - y²) + C
• ∫ tan^-1y dy = y tan^-1y - ½ ln |1+y²| + C
• ∫ cosec^-1y dy = y cosec^-1y + ln |y + √(y² - 1)| + C
• ∫ sec^-1y dy = y sec^-1y - ln |y + √(y² - 1)| + C
• ∫ cot^-1y dy = y cot^-1y + ½ ln |1+y²| + C

In addition to the above, other Integration Rules of Trigonometric Functions are mentioned as follows: 

• ∫1/√(1 - y²).dy = sin^-1y + C
• ∫ 1/(1 - y²).dy = - cos^-1y + C
• ∫ 1/y√(y² - 1).dy = sec^-1y + C
• ∫ 1/y√(y² - 1).dy = - cosec^-1 y + C
• ∫1/(1 + y²).dy = tan^-1y + C
• ∫ 1/(1 + y²).dx = - cot^-1y + C

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Integration Rules of Special Functions

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There are rules for integrating special kinds of rational functions in which denominators square themselves. The Integration Rules of Special Functions are mentioned as follows: 

• ∫1/ (y² - a²) dy = (1/2a) log|(y-a)/(y+a)| +C
• ∫1/ (a² - y²) dy = (1/2a) log|(a+y)/(a-y)| +C
• ∫ 1/ √(y² + a²) dy = log |y + √(y² + a²)|+C
• ∫1/ √(y² - a²) dy = log |y + √(y² - a²)|+C
• ∫1/ (a² + y²) dy = (1/a) tan ^-1 (y/a) + C
• ∫ 1/ √(a² - y²) dy = sin^-1 (y/a) +C

The Rules of Integration involving square roots are as follows:

• ∫1/ (y² - a²) dy = (1/2a) log|(y-a)/(y+a)| +C
• ∫1/ (a² - y²) dx = (1/2a) log|(a+y)/(a-y)| +C
• ∫ 1/ √(y² + a²) dy = log |y + √(y² + a²)|+C
• ∫1/ √(y² - a²) dy = log |y + √(y² - a²)|+C
• ∫1/ (a² + y²) dy = (1/a) tan ^-1 (y/a) + C
• ∫ 1/ √(a² - y²) dx = sin^-1 (y/a) +C

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ILATE Rule of Integration

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ILATE Rule is the Product Rule of Integration or the ‘Integration by Parts’ method. It is utilized for integrating the multiplied outcome of two functions. The ILATE Rule of Integration is denoted as 

However, when the functions are u and dv, the integration rule is applied by following the approach provided below:

• I: Inverse Trigonometric Function
• L: Logarithmic Function
• A:  Algebraic Function
• T: Trigonometric Function
• E: Exponential Function

The derivation of the rule is explained as follows:

∫ ln y dx = ∫ ln y · 1 dy

Here, the log function is y and the algebraic function is 1. Therefore, applying ILATE,

Let u = ln y and dv = 1. The outcome is 

du = (1/y) dy and v = ∫ 1 dy = y

Applying the ‘Integration by Parts’ method, 

∫ u dv = uv - ∫ v du

∫ ln y · 1 dy = (ln y) (y) - ∫ y (1/y) dy

∫ ln y dy = y ln y - ∫ 1 dy = y ln y - y + C

Read More: Double Integral

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Methods of Integration 

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Methods of Integration are mentioned as follows:

Integration by Parts

Product Rule of Integration is considered while multiplying two functions. It is also used while applying the ILATE Rule of Integration as discussed above. Splitting into partial functions is the key thing for the effective application of this method, after which the rule 1/(ay + b) dy = (1/a) ln |ay + b| + C is applied for integrating each of the partial functions. 

Integration by Substitution 

The method is applied when an integral provided is modified into an integral in its simple form through substitution of independent variables through others. The ‘‘Integration by Substitution’ is important as it helps in the effective application of the Integration Rules of Trigonometric Functions, the Integration Rules of Inverse Trigonometric Functions, and the Integration Rules of Special Functions. 

The method is explained as follows:

• Assumption of a part of integrand being v.
• Finding dv.
• Transforming the integral provided fully in terms of v
• Replacement of the value of v back in the outcome

Integration By Partial Fractions

In Integration by Partial Fractions, a rational function is split into partial fractions using a suitable rule and then integrated using the rule ∫ 1/(ax + b) dx = (1/a) ln |ax + b| + C to integrate each partial fraction.

Where x² – bx + c cannot be factorized further.

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Solved Examples on Integration Rules

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Here are a few solved examples on integration rules: 

Example 1: What is ∫ 8 a³ da?

Solution: We will take 8 out of integral,

∫ 8 a³ da = 8 ∫ a³ da

= 8 a⁴ / 4 + C

= 2 a⁴ + C

Example 2: What is ∫ 4 a³ da?

Solution: We will take 4 out of integral,

∫ 4 a³ da = 4 ∫ a³ da

= 4 a⁴ / 4 + C

= a⁴ + C

Example 3: What is ∫ (Cos a + a) da ?

Solution: ∫ (Cos a + a) da = ∫ Cos a da + ∫ a da

= sin a + a² /2 + C

Example 4: What is ∫ (Sin a + a) da ?

Solution: ∫ (Sin a + a) da = ∫ Sin a da + ∫ a da

= – Cos a + a² /2 + C

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Things to Remember

• Integration in calculus has led to the evolution of integral calculus. 
• Integration is vital for deriving various parameters such as area, volume, central point, and other types of quantities.
• Integration Rules are used for functions such as constants, variables, squares, reciprocal and exponential functions, and trigonometry. 
• The integration rules include Power Rule, Sum Rule, Difference Rule, Multiplication by Constant, and Product Rule. 
• Integration Rules are of utmost importance in terms of carrying out the integration of functions in an effective and efficient manner. 
• Integration Rules are used for the integration of complicated functions in order to derive the desired outcome without any major hindrances. 

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Previous Years’ Questions 

1. If \(∫xlog\,(1 + {1 \over x})\)= dx = f(x)log(x+1) + g(x)x² + Lx+C then... (BITSAT – 2017)
2. The area (in s units) of the region {(x,y) : x≥0, x + y≤3, x²≤4y and... (JEE Main – 2017)
3. For a>0, let the curves C_1​:y² = ax and C₂​:x² = ay intersect at origin O... (JEE Main – 2020)
4. The area of the region A=[(x,y) : 0≤y≤x∣x∣+1 and −1≤x≤1] in s units is... (JEE Main – 2019)
5. The area (in s units) bounded by the curves y... (COMEDK UGET - 2013)
6. Let the straight line x=b divide the area enclosed by... (AMUEEE - 2014)
7. The number of integral terms in the expansion…
8. If \({x^2 + 5} \over {(x^2 + 1) (x-2)} \)= \(A \over {x-2} \) + \(Bx + C \over {x^2 + 1}\), then A+B+C... (TS EAMCET - 2017)
9. You are given a curve, y=ln(x+e). What will be the area enclosed between... (JKCET - 2017)
10. The integral ∫cos (logx) dx is equal to… (JEE Main - 2019)