CPT Chapter Integral Calculus

CA CPT Exam  

With Anand.Duration:5 hrs 26mins

Intergration Duration (min:sec)
{modal https://www.youtube.com/embed/-EeUyEvk4EI?autoplay=1;rel=0|width=780|height=439|title=Introduction}Introduction{/modal} 08:36
{modal https://www.youtube.com/embed/mP2_XQjW_vc?autoplay=1;rel=0|width=780|height=439|title=Index}Index{/modal} 06:24
{modal https://www.youtube.com/embed/d5N4njlrGeQ?autoplay=1;rel=0|width=780|height=439|title=What is Integration}What is Integration{/modal} 03:23
{modal https://www.youtube.com/embed/inXZV4JUB7Q?autoplay=1;rel=0|width=780|height=439|title=Basic Formulae}Basic Formulae{/modal} 11:14
{modal https://www.youtube.com/embed/qjNg4FkSG7Q?autoplay=1;rel=0|width=780|height=439|title=Integration of Constant}Integration of Constant{/modal} 03:40
{modal https://www.youtube.com/embed/H8mtr8c2RB0?autoplay=1;rel=0|width=780|height=439|title=Example Basic Formulae}Example Basic Formulae{/modal} 06:06
{modal https://www.youtube.com/embed/ajHI_PH1lsE?autoplay=1;rel=0|width=780|height=439|title=Problems on Basic Formulae}Problems on Basic Formulae{/modal} 15:18
{modal https://www.youtube.com/embed/NV7vYDekeDY?autoplay=1;rel=0|width=780|height=439|title=Special Variable}Special Variable{/modal} 25:56
{modal https://www.youtube.com/embed/yVqLaEHPMyw?autoplay=1;rel=0|width=780|height=439|title=Method of Substitution & Examples}Method of Substitution & Examples{/modal} 15:07
{modal https://www.youtube.com/embed/pDHcexG6Svw?autoplay=1;rel=0|width=780|height=439|title=Problems on Method of Substitution}Problems on Method of Substitution{/modal} 26:29
{modal https://www.youtube.com/embed/IsMgMWgJsEY?autoplay=1;rel=0|width=780|height=439|title=Integration by Parts & Example}Integration by Parts & Example{/modal} 11:18
{modal https://www.youtube.com/embed/wxqZ-iyk17M?autoplay=1;rel=0|width=780|height=439|title=Problems Integration by Parts}Problems Integration by Parts{/modal} 19:29
{modal https://www.youtube.com/embed/GL44NPGvxF0?autoplay=1;rel=0|width=780|height=439|title=Special cases- Integration by Parts}Special cases- Integration by Parts{/modal} 06:17
{modal https://www.youtube.com/embed/LZl-4e55YsA?autoplay=1;rel=0|width=780|height=439|title=Partial Fraction}Partial Fraction{/modal} 08:52
{modal https://www.youtube.com/embed/ciDOn5JQYEY?autoplay=1;rel=0|width=780|height=439|title=Problem on Partial Fraction}Problem on Partial Fraction{/modal} 21:00
{modal https://www.youtube.com/embed/wFuW_H7GrUw?autoplay=1;rel=0|width=780|height=439|title=Special Substitution}Special Substitution{/modal} 28:07
{modal https://www.youtube.com/embed/EjJYakuzKvY?autoplay=1;rel=0|width=780|height=439|title=What is Definite Integration (DI)}What is Definite Integration (DI){/modal} 03:54
{modal https://www.youtube.com/embed/h6YyPm905Vk?autoplay=1;rel=0|width=780|height=439|title=Properties of DI}Properties of DI{/modal} 06:23
{modal https://www.youtube.com/embed/hAll5LsoK2Q?autoplay=1;rel=0|width=780|height=439|title=Examples of DI}Examples of DI{/modal} 06:14
{modal https://www.youtube.com/embed/cInjzCoaW9Y?autoplay=1;rel=0|width=780|height=439|title=Problems 1 DI}Problems 1 DI{/modal} 26:07
{modal https://www.youtube.com/embed/86vqD3nSoYA?autoplay=1;rel=0|width=780|height=439|title=Problems 2 DI}Problems 2 DI{/modal} 21:06
{modal https://www.youtube.com/embed/VMYYBQLkABI?autoplay=1;rel=0|width=780|height=439|title=Problems 3 DI}Problems 3 DI{/modal} 20:35
{modal https://www.youtube.com/embed/K6Dw_1cCDgA?autoplay=1;rel=0|width=780|height=439|title=Slope Problem}Slope Problem{/modal} 06:06
{modal https://www.youtube.com/embed/obS9tSRxRkE?autoplay=1;rel=0|width=780|height=439|title=Mis Problem}Mis Problem{/modal} 12:57
{modal https://www.youtube.com/embed/fsFeWan8azI?autoplay=1;rel=0|width=780|height=439|title=Summary}Summary{/modal} 05:28
Total 5:26:06

Few things which we have covered

What is Integration?
Basic Formulae
Special Variable
Actual Substitution
Integration by parts
Parts Special case
Partial Fraction
Special Substitution
Definite Integration
Properties of Definite Integration

What is Integration?
Integration is reverse of Differentiation
ex-

Symbol
∫x2dx

Integration of a constant-

∫ a dx =∫a.x0 dx

∫ 7 dx =

Addition & Subtraction to a constant-

3+k

3-k

Example-

∫(√x +1/√x)dx
(a) 2x1/2(1/3 x-1)
(b) 2x1/2(1/3 x+1)
(c) 2(1/3x + x1/2)
(d) None of these.

Evaluate = ∫5x2 dx :
(a) 5/3x3+ k (b) 5x3/3+k (c) 5x3 (d) none of these

Integration of 3 – 2x – x4 will become
(a) – x2 – x5 / 5 (b) 3x - x2 - x5 /5+ k (c) 3x -x2 +x5/5 +k (d) none of these

Given f(x) = 4x3 + 3x2 – 2x + 5 and ∫ f(x) dx is
(a) x4 + x3 – x2 + 5x (b) x4 + x3 – x2 + 5x + k
(c) 12x2 + 6x – 2x2 (d) none of these

Special Variables-

This is applied when by taking a linear function of x i.e ax or ax+b as X , we can apply any of the standard formulae

Here we apply standard formula and divide the answer with derivative of ax+b i.e a

Use method of substitution to integrate the function f(x)=(4x+5)6 and the answer is
(a) 1/28(4x+5)7+k
(b) (4x+5)7/7+k
(c) (4x+5)7/7
(d) none of these

Integrate (x+a)n and the result will be
(a) (x+a)n+1/n+1 + k
(b) (x+a)n+1/n+1
(c) (x+a)n+1
(d) none of these

Method of Substitution or Change of Variable-

Substitution
When we have a function & it’s derivatives
∫ (5x2+4x+3)4 (10x+4)dx

Note - Trick to identify which one is function & which is derivative

Evaluate: ∫dx/√(x2+a2)
(a) 1/2 - log (x+√x2+a2) + C
(b) log (x+√x2+a2) +C
(C) log (x √x2+a2)+ C
(d) 1/2log (x √x2+a2)+ C

Square Partial Integration-

Using method of partial fraction to evaluate
∫ (x+5) dx/(x +1) (x + 2)2 we get
(a) 4 log (x + 1) - 4log(x + 2)+3/x+2+k
(b) 4 log (x + 2) – 3/x +2)+k
(c) 4 log (x + 1) – 4 log(x+2)
(d) none of these

Special Substitution-

∫d( x2+1) / √x2+2 is equal to
(a) x/2 (√x2 +2)+k
(b) √x2 +2 +k
(c) 1/(x2+2)3/2 +k
(d) none of these

Definite Integration-

The value of ∫(1+logx)/x dx is:[Given Loge =1]
(a) 1/2
(b) 3/2
(c) 1
(d) 5/2

Summary-

What is Integration?
Basic Formulae
Special Variable
Actual Substitution
Integration by parts
Parts Special case
Partial Fraction
Special Substitution
Definite Integration
Properties of Definite Integration 


Notes

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