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Integration 1 11 TH A HL

The document provides a comprehensive list of fundamental integration formulae, detailing various integral calculations for different functions. It includes specific formulas for polynomials, exponential, logarithmic, and trigonometric functions, along with notes on applying these formulas to transformed variables. Additionally, it emphasizes the importance of adjusting for coefficients when substituting variables in integration.

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17 views11 pages

Integration 1 11 TH A HL

The document provides a comprehensive list of fundamental integration formulae, detailing various integral calculations for different functions. It includes specific formulas for polynomials, exponential, logarithmic, and trigonometric functions, along with notes on applying these formulas to transformed variables. Additionally, it emphasizes the importance of adjusting for coefficients when substituting variables in integration.

Uploaded by

shreyas.agrawal1002
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Q.

Ans.
Q.1

Ans.

Q.2
Ans.

Fundamental Integration Formulae .:


(1)
𝒙𝒏+𝟏 𝒅 𝒙𝒏+𝟏
(i) ∫ 𝒙𝒏 𝒅𝒙 = + 𝒄, 𝒏 ≠ −𝟏 ∵ ( 𝒏+𝟏 ) = 𝒙𝒏
𝒏+𝟏 𝒅𝒙
(ii) ∫ 𝒅𝒙 = 𝒙 + 𝒄
𝟏
(iii) ∫ 𝒅𝒙 = 𝟐√𝒙 + 𝒄,
√𝒙
𝟏 (𝒂𝒙+𝒃)𝒏+𝟏
(iv) ∫(𝒂𝒙 + 𝒃)𝒏 𝒅𝒙 = . +𝒄
𝒂 𝒏+𝟏
𝟏 𝒅 𝟏
(2) (i) ∫ 𝒅𝒙 = 𝒍𝒏 | 𝒙| + 𝒄 ∵ (𝒍𝒏 | 𝒙|) =
𝒙 𝒅𝒙 𝒙
𝟏 𝟏
(ii) ∫ 𝒅𝒙 = (𝒍𝒐𝒈 | 𝒂𝒙 + 𝒃| + 𝒄
𝒂𝒙+𝒃 𝒂
𝒅
(3) ∫ 𝒆𝒙 𝒅𝒙 = 𝒆𝒙 + 𝒄 ∵ (𝒆𝒙 ) = 𝒆𝒙
𝒅𝒙
𝒂𝒙 𝒅 𝒂𝒙
(4) ∫ 𝒂𝒙 𝒅𝒙 = +𝒄 ∵ ( ) = 𝒂𝒙
𝒍𝒏 𝒂 𝒅𝒙 𝒍𝒏 𝒂
𝒅
(5) ∫ 𝒔𝒊𝒏 𝒙 𝒅𝒙 = − 𝒄𝒐𝒔 𝒙 + 𝒄 ∵ (− 𝒄𝒐𝒔 𝒙) = 𝒔𝒊𝒏 𝒙
𝒅𝒙
𝒅
(6) ∫ 𝒄𝒐𝒔 𝒙 𝒅𝒙 = 𝒔𝒊𝒏 𝒙 + 𝒄 ∵ (𝒔𝒊𝒏 𝒙) = 𝒄𝒐𝒔 𝒙
𝒅𝒙
𝒅
(7) 𝒔𝒆𝒄𝟐 𝒙 𝒅𝒙 = 𝒕𝒂𝒏 𝒙 + 𝒄 ∵ (𝒕𝒂𝒏 𝒙) = 𝒔𝒆𝒄𝟐 𝒙
𝒅𝒙
𝒅
(8) ∫ 𝒄𝒐𝒔𝒆𝒄𝟐 𝒙 𝒅𝒙 = − 𝒄𝒐𝒕 𝒙 + 𝒄 ∵ (− 𝒄𝒐𝒕 𝒙) = 𝒄𝒐𝒔𝒆𝒄𝟐 𝒙
𝒅𝒙
𝒅
(9) ∫ 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙 𝒅𝒙 = 𝒔𝒆𝒄 𝒙 + 𝒄 ∵ (𝒔𝒆𝒄 𝒙) = 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙
𝒅𝒙
(10) ∫ 𝒄osec𝒙 𝒄𝒐𝒕 𝒙 𝒅𝒙 = −cosec𝒙 + 𝒄
𝒅
∵ (− 𝒄𝒐𝒔 𝒆 𝒄𝒙) = 𝒄𝒐𝒔 𝒆 𝒄𝒙 𝒄𝒐𝒕 𝒙
𝒅𝒙
(11) ∫ 𝒕𝒂𝒏 𝒙 𝒅𝒙 = − 𝒍𝒏 | 𝒄𝒐𝒔 𝒙 | + 𝒄 = 𝒍𝒏 𝒔𝒆𝒄 𝒙 | + 𝒄
𝒅
∵ (𝒍𝒏𝒄𝒐𝒔 𝒙) = − 𝒕𝒂𝒏 𝒙
𝒅𝒙
(12) ∫ 𝒄𝒐𝒕 𝒙 𝒅𝒙 = 𝒍𝒏 | 𝒔𝒊𝒏 𝒙 | + 𝒄 = − 𝒍𝒏 | cosec𝒙| + 𝒄
𝒅
∵ (𝒍𝒏𝒔𝒊𝒏 𝒙) = 𝒄𝒐𝒕 𝒙
𝒅𝒙
𝝅 𝒙
(13) ∫ 𝒔𝒆𝒄 𝒙 𝒅𝒙 = 𝒍𝒏 | 𝒔𝒆𝒄 𝒙 + 𝒕𝒂𝒏 𝒙 | + 𝒄 = 𝒍𝒏 𝒕𝒂𝒏 ( + ) + 𝒄
𝟒 𝟐
𝒅
∵ 𝒍𝒏 𝒔𝒆𝒄 𝒙 + 𝒕𝒂𝒏 𝒙) = 𝒔𝒆𝒄 𝒙
𝒅𝒙
𝒙
(14) cosec𝒙 𝒅𝒙 = 𝒍𝒏 | cosec𝒙 − 𝒄𝒐𝒕 𝒙 | + 𝒄 = 𝒍𝒏 𝒕𝒂𝒏 + 𝒄
𝟐
𝒅
∵ (𝒍𝒏 | cosec𝒙 − 𝒄𝒐𝒕 𝒙 |) = cosec𝒙
𝒅𝒙
𝒅𝒙
(15) ∫ = 𝒔𝒊𝒏−𝟏 𝒙 + 𝒄 = − 𝒄𝒐𝒔−𝟏 𝒙 + 𝒄
√𝟏−𝒙𝟐
𝒅 𝟏 𝒅 −𝟏
∵ (𝒔𝒊𝒏−𝟏 𝒙) = , (𝒄𝒐𝒔−𝟏 𝒙) =
𝒅𝒙 √𝟏−𝒙𝟐 𝒅𝒙 √𝟏−𝒙𝟐
𝒅𝒙 𝒙 𝒙
(16) ∫ = 𝒔𝒊𝒏−𝟏 + 𝒄 = − 𝒄𝒐𝒔−𝟏 + 𝒄
√𝒂𝟐 −𝒙𝟐 𝒂 𝒂
𝒅 𝒙 𝟏 𝒅 𝒙 −𝟏
∵
𝒅𝒙
(𝒔𝒊𝒏−𝟏 𝒂) = ,
𝒅𝒙
(𝒄𝒐𝒔−𝟏 ) =
𝒂
√𝒂𝟐 −𝒙𝟐 √𝒂𝟐 −𝒙𝟐
𝒅𝒙
(17) ∫ = 𝒕𝒂𝒏−𝟏 𝒙 + 𝒄 = − 𝒄𝒐𝒕−𝟏 𝒙 + 𝒄
𝟏+𝒙𝟐
𝒅 𝟏 𝒅 −𝟏
∵ (𝒕𝒂𝒏−𝟏 𝒙) = , (𝒄𝒐𝒕−𝟏 𝒙) =
𝒅𝒙 𝟏+𝒙𝟐 𝒅𝒙 𝟏+𝒙𝟐
𝒅𝒙 𝟏 −𝟏 𝒙 −𝟏 𝒙
(18) ∫ = 𝒕𝒂𝒏 +𝒄= 𝒄𝒐𝒕−𝟏 + 𝒄
𝒂𝟐 +𝒙𝟐 𝒂 𝒂 𝒂 𝒂
𝒅 −𝟏 𝒙 𝒂 𝒅 𝒙 −𝒂
∵
𝒅𝒙
(𝒕𝒂𝒏 𝒂
) =
𝒂𝟐 +𝒙𝟐 𝒅𝒙
(𝒄𝒐𝒕−𝟏 𝒂) = 𝒂𝟐+𝒙𝟐
𝒅𝒙
(19) ∫ = 𝒔𝒆𝒄−𝟏 𝒙 + 𝒄 = − 𝒄𝒐𝒔 ⥂ 𝒆𝒄−𝟏 𝒙 + 𝒄
𝒙√𝒙𝟐 −𝟏
𝒅 𝟏 𝒅 −𝟏
∵ (𝒔𝒆𝒄−𝟏 𝒙) = (cosec𝒙) =
𝒅𝒙 𝒙√𝒙𝟐 −𝟏 𝒅𝒙 𝒙√𝒙𝟐 −𝟏
𝒅𝒙 𝟏 𝒙 −𝟏 −𝟏 𝒙
(20) ∫ = 𝒔𝒆𝒄−𝟏 + 𝒄 = 𝒄𝒐𝒔𝒆𝒄 +𝒄
𝒙√𝒙𝟐 −𝒂𝟐 𝒂 𝒂 𝒂 𝒂
𝒅 𝒙 𝒂 𝒅 𝒙 −𝒂
∵
𝒅𝒙
(𝒔𝒆𝒄−𝟏 𝒂) = 𝒅𝒙
(𝒄𝒐𝒔𝒆𝒄−𝟏 ) =
𝒂
𝒙√𝒙𝟐 −𝒂𝟐 𝒙√𝒙𝟐 −𝒂𝟐

NOTE: In any of the fundamental integration formulae, if x is replaced


by ax + b, then the same formulae is applicable but we must
divide by coefficient of x or derivative of (ax + b) i.e., a. In
general,

𝟏
if ∫ 𝒇(𝒙)𝒅𝒙 = 𝝋(𝒙) + 𝒄, then∫ 𝒇(𝒂𝒙 + 𝒃)𝒅𝒙 = 𝝋(𝒂𝒙 + 𝒃) + 𝒄
𝒂
−𝟏
∫ 𝒔𝒊𝒏( 𝒂𝒙 + 𝒃)𝒅𝒙 = 𝒂
𝒄𝒐𝒔( 𝒂𝒙 + 𝒃) + 𝒄,
𝟏
∫ 𝒔𝒆𝒄( 𝒂𝒙 + 𝒃)𝒅𝒙 = 𝒂 𝒍𝒐𝒈 | 𝒔𝒆𝒄( 𝒂𝒙 + 𝒃) + 𝒕𝒂𝒏( 𝒂𝒙 + 𝒃)| + 𝒄 etc.

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