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Formulae of Integration

The document contains a comprehensive list of integration formulas, including basic integrals like ∫1 dx and ∫sin(x) dx, as well as more complex forms involving trigonometric and logarithmic functions. Each formula is presented with a general rule and includes a constant of integration, C. Additionally, it notes that these formulas remain valid if the variable x is replaced by a linear transformation ax + b, with the result divided by a.

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ke25092007
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29 views2 pages

Formulae of Integration

The document contains a comprehensive list of integration formulas, including basic integrals like ∫1 dx and ∫sin(x) dx, as well as more complex forms involving trigonometric and logarithmic functions. Each formula is presented with a general rule and includes a constant of integration, C. Additionally, it notes that these formulas remain valid if the variable x is replaced by a linear transformation ax + b, with the result divided by a.

Uploaded by

ke25092007
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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FORMULAE

1) ∫ 1 dx = 𝒙 + C

2) ∫ k dx = k 𝒙 + C , where k is a constant

𝒙𝒏+𝟏
3) ∫ 𝒙𝒏 dx = + C ; n ≠ -1
𝒏+𝟏

4) ∫ sin𝒙 dx = – cos𝒙 + C

5) ∫ cos𝒙 dx = sin𝒙 + C

6) ∫ sec2𝒙 dx = tan𝒙 + C

7) ∫ cosec2𝒙dx = – cot𝒙 + C

8) ∫ sec𝒙 tan𝒙 dx = sec 𝒙 + C

9) ∫ cosec𝒙 cot𝒙 dx = – cosec 𝒙 + C

𝟏
10) ∫ dx = log |𝒙| + C
𝒙

11) ∫ 𝒆𝒙 dx = 𝒆𝒙 + C

𝒂𝒙
12) ∫ 𝒂𝒙 dx = + C ; a>0, a≠1
𝒍𝒐𝒈𝒂

𝟏
13) ∫√ 𝒅𝒙 = 𝐬𝐢𝐧−𝟏 𝒙 + 𝑪 𝑶𝑹 − 𝐜𝐨𝐬−𝟏 𝒙 + 𝑪
𝟏−𝒙𝟐

𝟏
14) ∫ 𝟏+𝒙𝟐 𝒅𝒙 = 𝐭𝐚𝐧−𝟏 𝒙 + 𝑪 𝑶𝑹 − 𝐜𝐨𝐭 −𝟏 𝒙 + 𝑪

𝟏
15) ∫ 𝒅𝒙 = 𝐬𝐞𝐜 −𝟏 𝒙 + 𝑪 𝑶𝑹 − 𝐜𝐨𝐬𝐞𝐜 −𝟏 𝒙 + 𝑪
𝒙√𝒙𝟐 −𝟏
16) ∫ 𝒕𝒂𝒏𝒙 𝒅𝒙 = 𝐥𝐨𝐠|𝑺𝒆𝒄𝒙| + 𝑪

17) ∫ 𝑪𝒐𝒕𝒙 𝒅𝒙 = 𝐥𝐨𝐠|𝑺𝒊𝒏𝒙| + 𝑪

18) ∫ 𝑺𝒆𝒄𝒙 𝒅𝒙 = 𝐥𝐨𝐠|𝑺𝒆𝒄𝒙 + 𝒕𝒂𝒏𝒙| + 𝑪

19) ∫ 𝑪𝒐𝒔𝒆𝒄𝒙 𝒅𝒙 = 𝐥𝐨𝐠|𝑪𝒐𝒔𝒆𝒄𝒙 − 𝑪𝒐𝒕𝒙| + 𝑪

𝟏 𝟏 𝒙−𝒂
20) ∫ 𝒙𝟐−𝒂𝟐 𝒅𝒙 = 𝟐𝒂
𝒍𝒐𝒈 |
𝒙+𝒂
|+𝑪

𝟏 𝟏 𝒂+𝒙
21) ∫ 𝒂𝟐−𝒙𝟐 𝒅𝒙 = 𝟐𝒂
𝒍𝒐𝒈 |
𝒂−𝒙
|+𝑪

𝟏 𝟏 𝒙
22) ∫ 𝒙𝟐+𝒂𝟐 𝒅𝒙 = 𝐭𝐚𝐧−𝟏 ( ) + C
𝒂 𝒂

𝟏
23) ∫√ 𝒅𝒙 = 𝒍𝒐𝒈|𝒙 + √𝒙𝟐 − 𝒂𝟐 | + 𝑪
𝒙𝟐 −𝒂𝟐

𝟏
24) ∫√ 𝒅𝒙 = 𝒍𝒐𝒈|𝒙 + √𝒙𝟐 + 𝒂𝟐 | + 𝑪
𝒙𝟐 + 𝒂 𝟐

𝟏 𝒙
25) ∫√ 𝒅𝒙 = 𝐬𝐢𝐧−𝟏 ( ) + 𝑪
𝒂𝟐 −𝒙𝟐 𝒂

𝒙 𝒂𝟐
26) ∫ √𝒙𝟐 − 𝒂𝟐 𝒅𝒙 = √𝒙𝟐 − 𝒂𝟐 − 𝒍𝒐𝒈|𝒙 + √𝒙𝟐 − 𝒂𝟐 | + 𝑪
𝟐 𝟐

𝒙 𝒂𝟐
27) ∫ √𝒙𝟐 + 𝒂𝟐 𝒅𝒙 = 𝟐
√𝒙𝟐 + 𝒂𝟐 +
𝟐
𝒍𝒐𝒈|𝒙 + √𝒙𝟐 + 𝒂𝟐 | + 𝑪

𝒙 𝒂𝟐 𝒙
28) ∫ √𝒂𝟐 − 𝒙𝟐 𝒅𝒙 = √𝒂𝟐 − 𝒙𝟐 + 𝐬𝐢𝐧−𝟏 ( ) + 𝑪
𝟐 𝟐 𝒂

NOTE: In all the above formulae , if x is replaced by ax+b ( where a and b are constants) then all
the above formulae still remains true , provided the result on the RHS is divided by a
( ie the coefficient of x)

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