Question

how to use integration by parts to slove x(1+x)^6?

Answer 1

are you specifically asked to use integration by parts??
because its more easily solvable using substitution ...

Answer 2

u substitution is quicker :)

Answer 3

yes.. btw if you can plz tell how to use substitution also

Answer 4

To solve by u substitution the integral:

integral x * (1+x)^6 dx

u = 1 + x
du = dx

since u = 1 + x , solve for x
x = u - 1


integral x * (1+x)^6
= integral (u-1) * u^6 du
= integral (u*u^6 - u^6 ) du
= integral (u^7 - u^6)
= u^8/8 - u^7/7 + C
= (1+x)^8/8 - (1+x)^7/7 + C

Answer 5

so both ways come out the same

Answer 6

some fault

Answer 7

To solve by integration by parts the integral :
integral x * (1+x)^6 dx

u = x dv = (1+x)^6
du = dx v = (1+x)^7/7


integral udv = u v - integral v du


integral x * (1+x)^6
= x * ( 1 + x)^7/7 - integral( 1 + x )^7/7 dx
= x* ( 1 + x)^7/7 - (1 + x )^8/(7*8) + C

now its not obvious, but this is the same answer as using u sub

Answer 8

you have to factor the latter expression carefully

Answer 9

would you like me to?

Answer 10

the last answer is comming different

Answer 11

they are equivalent

Answer 12

the 6 is missing somewhere

Answer 13

thanks everyone

Answer 14

by the way
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Answer 15

oh sry @perl thats right i had some miscallculation

Answer 16

i'm having a bit of trouble proving that they are equivalent though :)

Answer 17

i plugged in random decimals and the output was equal

Answer 18

Claim:
x* ( 1 + x)^7/7 - (1 + x )^8/(7*8) = (1+x)^8/8 - (1+x)^7/7

Proof by algebra:

x* ( 1 + x)^7/7 - (1 + x )^8/(7*8)
= (1+x)^8/8 [ 1 - 1/(7(1+x)) ]