COMPLEX INTEGRATION QUESTIONS AND SOLUTIONS
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1. ∫ √
If u = tan , √ =√ u2 = tan2 , du = sec2 .d = (tan2 + 1)d = (u2 + 1)d
√ ( )
∫√ ∫ If u = x2, & u2 = x4 √ √ du = 2x.dx, Solve ∫ =∫
( ) ( ) ( ) ( ) ( ) ( )
∫* + =∫ ∫ = ∫ ∫
( ) ( )
( ) ( )
∫ ∫ What I did here is to simplify the powers of x.
( ) ( )
For the 1st integral (purple); y = , dy = ( )dx = ( )dx.
For the 2nd integral (blue); w = , dw = ( )dx = ( )dx
Substitute them for all the “x” term ∫ ∫ = ( ) ∫( *∫ ∫ +
(√ ) (√ ) √ √ √ )( √ ) √ √ √
√
Thus; ( ) [ | √ | | √ |] ( ) * | |+
√ √ √ √ √ √ √
√ √
Take y and w back to the „x‟ world: ( ) * | |+ ( ) * | |+
√ √ √ √ √ √ √ √
Recall that, x = √ , & x2 = : Now I‟ll take it to the „ ‟ world, & add the constant of integration
Youngscholar 1
You can message me for more solutions to questions on integral calculus via whatsapp on 08144345986. Or message me via my email
address
1. ∫ √
If u = tan , √ =√ u2 = tan2 , du = sec2 .d = (tan2 + 1)d = (u2 + 1)d
√ ( )
∫√ ∫ If u = x2, & u2 = x4 √ √ du = 2x.dx, Solve ∫ =∫
( ) ( ) ( ) ( ) ( ) ( )
∫* + =∫ ∫ = ∫ ∫
( ) ( )
( ) ( )
∫ ∫ What I did here is to simplify the powers of x.
( ) ( )
For the 1st integral (purple); y = , dy = ( )dx = ( )dx.
For the 2nd integral (blue); w = , dw = ( )dx = ( )dx
Substitute them for all the “x” term ∫ ∫ = ( ) ∫( *∫ ∫ +
(√ ) (√ ) √ √ √ )( √ ) √ √ √
√
Thus; ( ) [ | √ | | √ |] ( ) * | |+
√ √ √ √ √ √ √
√ √
Take y and w back to the „x‟ world: ( ) * | |+ ( ) * | |+
√ √ √ √ √ √ √ √
Recall that, x = √ , & x2 = : Now I‟ll take it to the „ ‟ world, & add the constant of integration
Youngscholar 1
