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Integral calculus
The document discusses various methods of integration in calculus, including techniques such as partial fractions, integration of rational algebraic functions, and integration of irrational functions. It outlines steps for performing these integrations and provides specific forms and properties associated with definite and indefinite integrals. Additionally, it mentions integration by parts and reduction formulas as important tools in calculus.
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Integral calculus
- 1. T H EI N T E G R A T I O N I S D E F I N E D A S T H E R E V E R S E P R O C E S S O F D I F F E R E N T I A T I O N A N D I S S O M E T I M E S T E R M E D A S A N T I - D E R I V A T I V E . T H U S I F ɸ ( 𝑥 ) I S D I F F E R E N T I A L F U N C T I O N O F X S U C H T H A T 𝑑 𝑑 𝑥 𝜙 𝑥 T H E N ɸ ( 𝑥 ) I S C A L L E D A N T I - D E R I V A T I V E O R P R I M I T I V E O R A N I N D E F I N I T E I N T E G R A L O R S I M P L Y A N I N T E G R A L O F F ( X ) . Integral Calculus
- 2. Methods of integration Method 1. Integrals of functions of the form 𝒇 𝒙 𝒏 𝒇′ 𝒙 𝒅𝒙 = 𝒇 𝒙 𝒏+𝟏 𝒏+𝟏 Note : when n = -1 , 𝑓′ 𝑥 𝑓 𝑥 𝑑𝑥 = 𝑙𝑜𝑔 𝑓(𝑥)
- 3. Method 2.Evaluate Integral by using partial fraction method If the denominator can be resolved into rational factors, the method of partial fraction is to be used Type 1. 𝒅𝒙 𝒂 𝟏 𝒙+𝒃 𝟏 𝒂 𝟐 𝒙+𝒃 𝟐 Step 1. Take 𝟏 𝒂 𝟏 𝒙+𝒃 𝟏 𝒂 𝟐 𝒙+𝒃 𝟐 = 𝑨 𝒂 𝟏 𝒙+𝒃 𝟏 + 𝑩 𝒂 𝟐 𝒙+𝒃 𝟐 Step 2. Find the value of A and B from step 1 Step 3. Substitute the value of A and B and then integrate Type 2. 𝒅𝒙 𝒂 𝟏 𝒙+𝒃 𝟏 𝟐 𝒂 𝟐 𝒙+𝒃 𝟐 Step 1. Take 𝟏 𝒂 𝟏 𝒙+𝒃 𝟏 𝟐 𝒂 𝟐 𝒙+𝒃 𝟐 = 𝑨 𝒂 𝟏 𝒙+𝒃 𝟏 + 𝑩 𝒂 𝟏 𝒙+𝒃 𝟏 𝟐 + 𝑪 𝒂 𝟐 𝒙+𝒃 𝟐 Step 2. Find the value of A,B and C from step 1 Step 3. Substitute the value of A,B and C and then integrate Type 3. 𝒅𝒙 (𝒂 𝟏 𝒙+𝒃 𝟏)(𝒂 𝟐 𝒙 𝟐+𝒃 𝟐 𝒙+𝒄 𝟐 ) Step 1. Take 𝟏 𝒂 𝟏 𝒙+𝒃 𝟏 𝒂 𝟐 𝒙 𝟐+𝒃 𝟐 𝒙+𝒄 𝟐 = 𝑨 𝒂 𝟏 𝒙+𝒃 𝟏 + 𝑩𝒙+𝒄 𝒂 𝟐 𝒙 𝟐+𝒃 𝟐 𝒙+𝒄 𝟐 Step 2. Find the value of A,B and C from step 1 Step 3. Substitute the value of A,B and C and then integrate
- 4. Integration of rationalalgebraic functions Method 3. Integration of the form 𝒂 𝟏 𝒙 𝟑+𝒃 𝟏 𝒙 𝟐+𝒄 𝟏 𝒙+𝒅 𝟏 𝒂 𝟐 𝒙 𝟐+𝒃 𝟐 𝒙+𝒄 𝟐 𝒅𝒙 If the degree of the numerator is equal or greater than the degree of the denominator, divide the numerator by the denominator and then integrate
- 5. Method 4.Integration of the form 𝒅𝒙 𝒂𝒙 𝟐+𝒃𝒙+𝒄 If the denominator is of second degree and does not resolve into rational factors. Then change the denominator into complete square form, the integral reduces to one of the following three forms 1. 𝑑𝑥 𝑥2+𝑎2 = 1 𝑎 tan−1 𝑥 𝑎 2. 𝑑𝑥 𝑥2−𝑎2 = 1 2𝑎 log 𝑥−𝑎 𝑥+𝑎 3. 𝑑𝑥 𝑎2−𝑥2 = 1 2𝑎 log 𝑎+𝑥 𝑎−𝑥 Method 5. Integration of the form 𝒑𝒙+𝒒 𝒅𝒙 𝒂𝒙 𝟐+𝒃𝒙+𝒄 Step 1.Put 𝑝𝑥 + 𝑞 = 𝐴 𝑑 𝑑𝑥 𝑎𝑥2 + 𝑏𝑥 + 𝑐 + 𝐵 Step 2. Find the value of A and B Step 3. Then given integral becomes 𝑝𝑥+𝑞 𝑑𝑥 𝑎𝑥2+𝑏𝑥+𝑐 = 𝐴 log(𝑎𝑥2 + 𝑏𝑥 + 𝑐) + 𝐵 𝑑𝑥 𝑎𝑥2+𝑏𝑥+𝑐 Step 4. Substitute the values of A and B and use method 4 to solve 2nd integration.
- 6. Integration of irrationalfunctions Method 6. Integration of the form 𝒅𝒙 𝒂𝒙 𝟐+𝒃𝒙+𝒄 Change the denominator into complete square form, the integral reduces to one of the following three forms 1. 𝑑𝑥 𝑎2−𝑥2 = sin−1 𝑥 𝑎 2. 𝑑𝑥 𝑎2+𝑥2 = sinh−1 ( 𝑥 𝑎 ) 3. 𝑑𝑥 𝑥2−𝑎2 = cosh−1 𝑥 𝑎 Method 7. Integration of the form (𝒑𝒙+𝒒)𝒅𝒙 𝒂𝒙 𝟐+𝒃𝒙+𝒄 Step 1.Put 𝑝𝑥 + 𝑞 = 𝐴 𝑑 𝑑𝑥 𝑎𝑥2 + 𝑏𝑥 + 𝑐 + 𝐵 Step 2. Find the value of A and B Step 3. Then given integral becomes 𝑝𝑥+𝑞 𝑑𝑥 𝑎𝑥2+𝑏𝑥+𝑐 = 𝐴 𝑎𝑥2 + 𝑏𝑥 + 𝑐 + 𝐵 𝒅𝒙 𝒂𝒙 𝟐+𝒃𝒙+𝒄 Step 4. Substitute the values of A and B and use method 6 to solve 2nd integration Method 8. Integration of the form 𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄 𝒅𝒙 Change the function inside the square root into complete square form, the integral reduces to one of the following three forms 1. 𝑎2 − 𝑥2 𝑑𝑥 = 1 2 𝑥 𝑎2 − 𝑥2 + 1 2 𝑎2 sin−1 𝑥 𝑎 2. 𝑥2 + 𝑎2 𝑑𝑥 = 1 2 𝑥 𝑥2 + 𝑎2 + 1 2 𝑎2 sinh−1 𝑥 𝑎 3. 𝑥2 − 𝑎2 𝑑𝑥 = 1 2 𝑥 𝑥2 − 𝑎2 − 1 2 𝑎2 cosh−1 𝑥 𝑎 Method 9. Integration of the form 𝒅𝒙 𝒑𝒙+𝒒 𝒂𝒙 𝟐+𝒃𝒙+𝒄 Step 1. Put 𝑝𝑥 + 𝑞 = 1 𝑡 Step 2. Then the given integration reduced to method 6 and solve
- 7. Definite Integrals Suppose𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝜙 𝑥 𝑎 𝑏 = 𝜙 𝑏 − 𝜙(𝑎) Here, a is called the lower limit and b is called the upper limit of the definite integral 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
- 8. Properties ofDefinite Integrals 1. 𝑎 𝑏 [ƒ(𝑥) + 𝑔(𝑥) 𝑑𝑥 = 𝑎 𝑏 ƒ(𝑥) 𝑑𝑥 + 𝑎 𝑏 𝑔(𝑥) 𝑑𝑥 2. 𝑎 𝑏 𝑘ƒ(𝑥) 𝑑𝑥 = 𝑘 𝑎 𝑏 ƒ(𝑥) 𝑑𝑥 3. 𝑎 𝑎 ƒ(𝑥) 𝑑𝑥 = 0 4. 𝑎 𝑏 ƒ(𝑥) 𝑑𝑥 = − 𝑏 𝑎 ƒ(𝑥) 𝑑𝑥 5. 𝑎 𝑏 ƒ(𝑥) 𝑑𝑥 + 𝑏 𝑐 ƒ(𝑥) 𝑑𝑥 = 𝑎 𝑐 ƒ(𝑥) 𝑑𝑥, where 𝑎 < 𝑐 < 𝑏 6. 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑧 𝑑𝑧 = 𝑎 𝑏 𝑓 𝑡 𝑑𝑡
- 9. The definiteintegral is independent of variable used. 7. 0 𝑎 𝑓 𝑥 𝑑𝑥 = 0 𝑎 𝑓 𝑎 − 𝑥 𝑑𝑥 8. −𝑎 𝑎 𝑓 𝑥 𝑑𝑥 = 0 , 𝑖𝑓 𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑖. 𝑒 𝑓 −𝑥 = −𝑓(𝑥) 2 0 𝑎 𝑓 𝑥 𝑑𝑥, 𝑖𝑓 𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, 𝑖. 𝑒 𝑓 −𝑥 = (𝑓(𝑥) 9. 0 𝑛𝑎 𝑓 𝑥 𝑑𝑥 = 𝑛 0 𝑎 𝑓 𝑥 𝑑𝑥 , 𝑖𝑓 𝑓 𝑎 + 𝑥 = 𝑓 𝑥 10. 0 𝑎 𝑓 𝑥 𝑑𝑥 = 0, 𝑖𝑓 𝑓 𝑎 − 𝑥 = −𝑓(𝑥) 2 0 𝑎 2 𝑓 𝑥 𝑑𝑥, 𝑖𝑓 𝑓 𝑎 − 𝑥 = 𝑓(𝑥)
- 10. Integration byparts If u and v are any two integrable functions, then 𝑢 𝑑𝑣 = 𝑢𝑣 − 𝑣 𝑑𝑢 Note: To choose u → I LATE I-Inverse Trigonometric L-Logarithm A-Algebraic T-Trigonometric E-Exponential Bernoulli’s formula 𝑢𝑑𝑣 = 𝑢𝑣 − 𝑢′ 𝑣1 + 𝑢′′ 𝑣2 + ⋯ Where 𝑢′ = 𝑑𝑢 𝑑𝑥 , 𝑢′′ = 𝑑2 𝑢 𝑑𝑥2 , … . 𝑣1 = 𝑣 𝑑𝑥 , 𝑣2 = 𝑣1 𝑑𝑥 , …
- 11. Reduction formula 1. 0 𝜋 2 𝑆𝑖𝑛 𝑛 𝑥 𝑑𝑥 = 𝑛−1 𝑛 𝑛−3 𝑛−2 … 1 2 𝜋 2 , 𝑖𝑓 𝑛 𝑠 𝑒𝑣𝑒𝑛 = 𝑛−1 𝑛 𝑛−3 𝑛−2 … 2 3 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 2. 0 𝜋 2 cos 𝑛 𝑥 𝑑𝑥 = 𝑛−1 𝑛 𝑛−3 𝑛−2 … 1 2 𝜋 2 , 𝑖𝑓 𝑛 𝑠 𝑒𝑣𝑒𝑛 = 𝑛−1 𝑛 𝑛−3 𝑛−2 … 2 3 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑