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UNIT- 1
INTEGRATION
Define an integral A function is called an anti derivative or integral of a function on an interval if for every value of in (i.e) If the derivative of a function w.r.to ,then we say that the integral of w.r.to is. (i.e) (^) ∫
Evaluate (^) ∫.
Solution:
Evaluate: (^) ∫ √
Solution:
Evaluate: (^) ∫
Solution: We know that,
[ ]
∫ ∫ [^ ]
∫[ ]
[∫ ∫ ]
[ ]
Evaluate: (^) ∫ √.
Solution:
∫ √ ∫ √^ [ ]
Evaluate: (^) ∫ √.
Solution: We know that,
Integrate: (^) ∫.
Solution:
We know that,
Integrate: (^) ∫.
Solution:
We know that,
Evaluate (^) ∫
Solution:
Evaluate ∫
Solution: We know that,
[ ]
[ ]
[ ]
∫ ∫[ ]
[ ]
[ ]
Integration by Method of substitution
Integrate: (^) ∫.
Solution:
Integrate: (^) ∫.
Solution:
Integrate: (^) ∫.
Solution:
Integrate: (^) ∫.
Solution:
Evaluate: (^) ∫.
Solution:
Evaluate (^) ∫.
Solution:
∫ ∫
Put
Evaluate (^) ∫
Solution:
Evaluate (^) ∫ d
Solution:
Evaluate ∫
Solution:
Evaluate (^) ∫.
Solution:
Evaluate (^) ∫
Solution:
Evaluate (^) ∫.
Solution: and
Solution:
∫ √^ ∫ √
Let
∫ √^ ∫ (√^ ) ( *
[ , √^ ( )-]
[ √^ ( *]
Integration by parts Let u & v be two functions of x then ∫ ∫
Evaluate (^) ∫ 3.
Solution:
∫ [^ ∫ ]
[ ] [ ]
Evaluate ∫. Solution: Applying integration by parts, Let
Again applying integration by parts
Let
q ⇒
Evaluate∫. Solution:
Use Integration by parts method
√
∫ ∫. / √ ∫ √ ∫ √ ∫ √
∫ [
√ ]
6 √ 7 [√ ]
q
∫ [ √^ ]
[ ( *]
[( ) ( ( ) )]
[( ) ]
( ) [ ]
Evaluate: (^) ∫.
Solution: We know that,
∫ [ ]
Evaluate: (^) ∫.
Solution:
∫ ∫ [ ]
[ ]
Evaluate
2
1
lo g^ x d x.
Solution:
∫ ∫
∫ [^ ]^ ∫
[ ] [ ] [ ]
Evaluate
4
0
ta n x d x
Solution: We know that,
∫ [ ]
[ (
* ]
[ √ ]
Evaluate:
a x a
e d x
Solution: We know that,
∫ [^ ]
[ ]
Evaluate (^) ∫.
Solution:
[ ]
[ ( * ] [ ]
[ ] ( *
Evaluate (^) ∫.
Solution: We know that,
[ ]
∫ ∫ [ ]
∫[ ]
[ ]
[ ( ) ]
[ ] [ ( ) ]
Evaluate (^) ∫ √.
Solution:
∫ √ ∫ 6 7 6 7 [ ]
Evaluate (^) ∫.
Solution:
Evaluate∫ √
Solution:
√
∫ √^ ∫
√ ∫
√
[
√ ∫
√ ]
*√ √^ [^ ]
√
[√ √^ * √^ +]
Evaluate∫.
Solution:
∫
L
∫ [ ]
[ ]
Evaluate (^) ∫.
Solution:
Evaluate∫.
Solution: Use Integration by parts method Let
∫ [^ ]^ ∫
Again Using Integration by parts method, Let
∫ [ ∫ ]
Evaluate∫.
Solution: Use Integration by parts method ,
∫ [ ] ∫
