Integration

<Integration by parts>

Solve ∫ t sin t dt. 

We can easily solve it with tabular integration.

 

[derived]  [interated]

t                 sin t

1                 -cos t

0                -sin t

 

Point: 

1. Functions derived to make them simple to the left

2. Functions integrated to make them simple to the right

 

Then pairing t and -cos t, 1 and -sin t, and 0 and -sin t,

and multiplying each, we get

-t cos t, -sin t, and 0,

 

Multiplying +1 and -1 in this order by each.

 

This results in

∫ t sint dt = -t cos t + sin t + C, where C is a constant number.

 

<Example>

∫x^3 exp(x^2)dx = ∫x^2 exp(x^2) (xdx) = ∫x^2 exp(x^2) (1/2dx^2)

Let t = x^2,

[derived]  [interated]

t                 exp(t)

1                 exp(t)

0                exp(t)

 

∫t exp(t) (1/2dt) = 1/2 {t exp(t) - exp(t)} + C

Therefore

∫x^3 exp(x^2)dx = 1/2 {x^2 exp(x^2) - exp(x^2)} + C