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Integration by Parts

Integration by Parts

A formula used to integrate the product of two functions.

 

Formula: Integration by parts formula: the integral of u dv equals uv minus the integral of v du
   
Example 1: Evaluate Integral of x times e to the power of x/2, dx: ∫ xe^(x/2) dx.
 

Use u = x and dv = ex/2 dx. Then we get du = dx and v = 2ex/2. This can be summarized:

u = x dv = ex/2 dx
du = dx v = 2ex/2

It follows that

Two-line integral solution: ∫xe^(x/2)dx = 2xe^(x/2) − ∫2e^(x/2)dx = 2xe^(x/2) − 4e^(x/2) + C

 
Example 2: Evaluate Integral of tan⁻¹(x) dx.
 

 

Use the following:

u = tan-1 x dv = dx
du = 1 divided by (1 + x squared), times dx; substitution formula for derivative of arctan(x) v = x

Thus

Step-by-step integration of arctan(x): x·tan⁻¹x − ½∫2x/(1+x²)dx = x·tan⁻¹x − ½ln|1+x²| + C

 
Example 3: Evaluate Integral of e^x times sin x dx.
 

 

Let I =Integral of e^x times sin x dx. Proceed as follows:

u = sin x dv = ex dx
du = cos x dx v = ex

Thus

Integral of e^x sin x dx = e^x sin x − integral of e^x cos x dx

Now use integration by parts on the remaining integral . Use the following assignments:

u = cos x dv = ex dx
du = –sin x dx v = ex

Thus

Two-step integration by parts derivation: ∫eˣsin x dx = eˣsin x − eˣcos x − ∫eˣsin x dx

Note that Integral of e^x times sin x dx appears on both sides of this equation. Replace it with I and then solve.

Three-step algebraic derivation solving for I: I = e^x·sin x − e^x·cos x − I; 2I = e^x·sin x − e^x·cos x; I = ½e^x·sin x −...

We finally obtain

The integral of e^x sin x dx equals (1/2)e^x sin x minus (1/2)e^x cos x plus C

 

See also

Integration methods