Reduction Formula in Integration: Types and Solved Example

Reduction formula important method of integration. It is effective in solving difficult and complex integration questions. It is used for various functions including trigonometric, logarithmic, exponential, and many others. To simplify the integration of these functions, the reduction formula comes into existence. These reduction formulas tend to calculate the solution of integral complex problems and help us to reduce the integral degree and evaluate the integrals in a limited number of steps. The reduction formula follows the rules of integration and is derived from the formulas of integration.

Read More: Applications of the Integrals

Key Terms: Integrals, Integration, Trigonometric, Logarithmic, Exponential, Algebraic Functions, Inverse Trigonometric Function

---

Formulas for Reduction in Integration

[Click Here for Sample Questions]

There are several types of reduction formulas that are required to solve complex problems related to different functions. The different types of reduction formulas include: Reduction Formula for Exponential Functions, Reduction Formula for Inverse Trigonometric Functions, Reduction Formulas For Trigonometric Functions, Reduction Formula for Algebraic Functions, Reduction Formula For Trigonometric Functions, Reduction Formula for Logarithmic Functions, and Reduction Formula for Hyperbolic Trigonometric Functions

Integrals Detailed Video Explanation

Read More: Integrals

---

Reduction Formula for Exponential Functions

[Click Here for Previous Year Questions]

• ∫ yⁿ e^my dy = 1/myⁿe^my –n/m y^n-1 e^mydy
• ∫ dy / sinhⁿy = -1/n sinh^n-1 y cosh y – (n-1/n) ∫ sinh ^n-2 y dy
• ∫ e^my / yⁿ dy = e^my (n-1)y^n-1 + m/n-1 ∫ e^mx/x^n-1 y
• ∫ Sinhⁿ ydy = -1/n Sin h^n-1 cosh y – n-1/n ∫ Sinh^n-2 y dy

Read More: 

---

Reduction Formula for Inverse Trigonometric Functions

[Click Here for Sample Questions]

• ∫ yⁿ arcsin y dy = yⁿ⁺¹/ n+1 arcsin y -1/n+1 ∫ y^n-1/\( \sqrt{} \) 1-y2dy
• ∫ yⁿ arccos y dy = yⁿ⁺¹/ n+1 arc cos y -1/n+1 ∫ y^n-1\( \sqrt{} \) 1-y2dy
• ∫ yⁿ arctan y dy = yⁿ⁺¹/ n+1 arctan y -1/n+1 ∫ y^n-1\( \sqrt{} \) 1-y2dy

Read More: Vectors

---

Reduction Formula For Trigonometric Functions

[Click Here for Previous Year Questions]

• ∫ Sinⁿ(y) dy = -sin^n-1 (y) cos (y)/n + n-1/n Sin^n-2 (y) dy
• ∫ yⁿ Sinⁿ(y) dy = -yⁿcos (y) + n ∫ y^n-1 cos (y) dy
• ∫ yⁿ Cos(y) dy = yⁿSin (y) - n ∫ y^n-1 sin (y) dy
• ∫ tanⁿ (y) dy = - tan^n-1 (y)/n-1 – ∫ tan^n-2 (y) dy
• ∫ Sinⁿ(y) dy Cos^m(y) dy = sinⁿ⁺¹ (y) cos^m-1(y)/n+m + m-1/n+m ∫ Sinⁿ(y) Cos^m-2(y)dy
• ∫ \(\frac{dx}{sin^n x} \) = \(\frac{cos x}{(n-1) sin^{n-1} x} + \frac{(n-2)}{(n-1)}\)∫ \(\frac{dx}{sin^{n-2} x'} \) n ≠ 1
• ∫ \(\frac{dx}{cos^n x} \) = \(\frac{sin x}{(n-1) cos^{n-1} x} + \frac{(n-2)}{(n-1)}\)∫ \(\frac{dx}{cos^{n-2} x'} \) n ≠ 1

Read More: Integration by Parts

---

Reduction Formula for Algebraic Functions

[Click Here for Sample Questions]

• ∫ yⁿ / ayⁿ + b dy = y/a-b/a ∫ dy/ayⁿ + b
• ∫ dy (ay² + by + C)ⁿ = - 2ay- b/ (n-1)(b² – 4ac) (ay² + by + C)^n-1 -2(2n-3)a/(n-1)(b² – 4ac) ∫ dy/(ay² + by + C)^n-1 , n ≠ 1
• ∫ dy/(y²- a²)ⁿ = x/2(n-1)a²(y² – a²)^n-1 – 2n-3/2(n-1)a² ∫ dy(y² – a²)^n-1 , n ≠ 1
• ∫ dy/(y² + a²)ⁿ = x/2(n-1)a²(y² – a²)^n-1 + 2n-3/2(n-1)a² ∫ dy(y² – a²)^n-1 , n ≠ 1

Read More: Difference Between Variance and Standard Deviation

---

Reduction Formula For Trigonometric Functions

[Click Here for Previous Year Questions]

• ∫ Inⁿ y dy = yⁿ⁺¹ In^m y/ n +1 xⁿ - m/n+1 ∫ xⁿIn^m-1 y dy
• ∫ In^m y/yⁿ = In^my + (n-1)y^{n +1} + m/n-1 ∫ In^m-1y/yⁿ dy

Read More: Continuous Integration

---

Reduction Formula for Logarithmic Functions

[Click Here for Sample Questions]

• ∫ xⁿ In^m dx = \(\frac{x^{n+1}ln^mx}{n+1} - \frac{m}{n+1}\)∫ xⁿ In^{m – 1}xdx
• ∫ \(\frac{ln^mx}{x^n} = - \frac{ln^mx}{(n-1)x^{n+1}} + \frac{m}{n-1} \) = ∫ \(\frac{ln^{m-1}x}{x^n}\), n ≠ 1

Read More: u substitution formula

---

Reduction Formula for Hyperbolic Trigonometric Functions

[Click Here for Previous Year Questions]

• ∫sinhⁿx dx = −(1/n) sinhⁿ⁻¹ x cosh x − (n−1/n) ∫sinhⁿ⁻²x dx
• ∫coshⁿx dx = (1/n) sinh x coshⁿ⁻¹ x − (n−1/n) ∫coshⁿ⁻²x dx
• ∫tanhⁿx dx = −(1/n−1) tanhⁿ⁻¹x + ∫tanhⁿ⁻²x dx, n≠1
• ∫cothⁿx dx = −(1/n−1)cothⁿ⁻¹x + ∫cothⁿ⁻²x dx, n≠1
• ∫ \(\frac{dx}{sinh^nx} = - \frac{cosh x}{(n-1) sinh^{n-1}x} + \frac{(n-2)}{(n-1)}\)∫ \(\frac{dx}{sinh^{n-2}x'}\), n≠1
• ∫ \(\frac{dx}{cosh^nx} = - \frac{sinh x}{(n-1) cosh^{n-1}x} + \frac{(n-2)}{(n-1)}\)∫ \(\frac{dx}{cosh^{n-2}x'}\), n≠1
• ∫sinhⁿxcosh^mxdx = \(\frac{sinh^{n+1}xcosh^{m-1}x}{n+m} + \frac{(m-1)}{(n+m)}\)∫ sinhⁿxcosh^m-2xdx
• ∫sechⁿxdx = \(\frac{sech^{n-2}xtanhx}{n-1} + \frac{n-2}{n-1}\)∫sech^n-2xdx,  n≠1

Read More:

---

Things to Remember

• Reduction formulas tend to calculate the solution of integral complex problems and help us to reduce the integral degree and evaluate the integrals in a limited number of steps.
• Integration is a method to calculate the antiderivative of a given function.
• The different types of categories of reduction formula include the reduction formula for trigonometric functions, inverse trigonometric functions, exponential functions, logarithmic functions, algebraic functions, and hyperbolic trigonometric functions. 
• ∫ yⁿ e^my dy = 1/myⁿe^my –n/m y^n-1 e^mydy
• ∫sinhⁿx dx = −(1/n) sinhⁿ⁻¹ x cosh x − (n−1/n) ∫sinhⁿ⁻²x dx
• ∫xⁿ ln^mx dx = \(\frac{x^{n+1}ln^{m}x}{n+1}\) - \(\frac{m}{n+1}\) ∫xⁿln^m−1xdx
• ∫ Sinⁿ(y) dy = -sin^n-1 (y) cos (y)/n + n-1/n Sin^n-2 (y) dy

Read More: Three-Dimensional Geometry Introduction

---

Previous Year Questions

1. The area enclosed between this curve and the coordinate axes… [JKCET – 2017]
2. The area (in sq units) of the region described by… [JEE Main – 2014]
3. The area (in s units) of the quadrilateral formed by the tangents… [JEE Main – 2015]
4. The area (in s units) bounded by the curve y… [COMEDK UGET – 2013]
5. If the line x = b bisects the area bounded by the curves, C₁​ and C₂… [JEE Main – 2020]
6. The greatest integer less than or equal to “t”…  [JEE Main – 2019]
7. [x] denotes the greatest integer less than ²⁰C_r​ … [JEE Main – 2019]
8. The value of the integral… [UPSEE – 2018]
9.  \(\int {x^3 - 1}\over {x^3 - x}\)dx… [COMEDK UGET – 2015]
10. The only integral root of the equation … [AMUEEE – 2016]