{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

int_by_parts

# int_by_parts - Harvey Mudd College Math Tutorial...

This preview shows pages 1–2. Sign up to view the full content.

Harvey Mudd College Math Tutorial: Integration by Parts We will use the Product Rule for derivatives to derive a powerful integration formula: Start with ( f ( x ) g ( x )) 0 = f ( x ) g 0 ( x ) + f 0 ( x ) g ( x ). Integrate both sides to get f ( x ) g ( x ) = Z f ( x ) g 0 ( x ) dx + Z f 0 ( x ) g ( x ) dx . (We need not include a constant of integration on the left, since the integrals on the right will also have integration constants.) Solve for Z f ( x ) g 0 ( x ) dx , obtaining Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z f 0 ( x ) g ( x ) dx. This formula frequently allows us to compute a difficult integral by computing a much simpler integral. We often express the Integration by Parts formula as follows: Let u = f ( x ) dv = g 0 ( x ) dx du = f 0 ( x ) dx v = g ( x ) Then the formula becomes Z u dv = uv - Z v du. To integrate by parts, strategically choose u , dv and then apply the formula. Example Let’s evaluate Z xe x dx . Let u = x dv = e x dx du = dx v = e x Then by integration by parts, Z xe x = xe x - Z e x dx = xe x - e x + C. A Faulty Choice A Reduction Formula Integration by parts “works” on definite integrals as well: Z b a u dv =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

int_by_parts - Harvey Mudd College Math Tutorial...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online