Integration by Parts
- is an integration technique by expanding the differential of a product of functions
𝑑(𝑢·𝑣)and expressing the original integral in terms of a known integral∫𝑣·𝑑𝑢
Definition/Intuition
Starts with product rule:
𝑑(𝑢·𝑣) = 𝑢·𝑑𝑣 + 𝑣·𝑑𝑢
Next, integrate both sides:
∫𝑑(𝑢·𝑣) = ∫𝑢·𝑑𝑣 + ∫𝑣·𝑑𝑢𝑢·𝑣 = ∫𝑢·𝑑𝑣 + ∫𝑣·𝑑𝑢
Rearranging yields:
∫𝑢·𝑑𝑣 = 𝑢·𝑣 - ∫𝑣·𝑑𝑢
Examples
∫𝑥·𝑐𝑜𝑠(𝑥)·𝑑𝑥
consider integral
∫𝑥·𝑐𝑜𝑠(𝑥)·𝑑𝑥where:
𝑢 = 𝑥
𝑑𝑣 = 𝑐𝑜𝑠(𝑥)·𝑑𝑥
𝑑𝑢 = 𝑑𝑥
𝑣 = 𝑠𝑖𝑛(𝑥)so integration by parts gives:
∫𝑥·𝑐𝑜𝑠(𝑥)·𝑑𝑥 = 𝑥·𝑠𝑖𝑛(𝑥) - ∫𝑠𝑖𝑛(𝑥)·𝑑𝑥
∫𝑥·𝑐𝑜𝑠(𝑥)·𝑑𝑥 = 𝑥·𝑠𝑖𝑛(𝑥) - [-𝑐𝑜𝑠(𝑥) ± 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡]
∫𝑥·𝑐𝑜𝑠(𝑥)·𝑑𝑥 = 𝑥·𝑠𝑖𝑛(𝑥) + 𝑐𝑜𝑠(𝑥) ± 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡