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Wize University Calculus 2 Textbook > Integration Techniques

Integration by Parts

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Integration by Parts (I by P)

Identifying Clues

There is a product of two different function types. ("reverse product rule).
  • ∫01xe5x dx\int_0^1xe^{5x}\ dx
  • ∫x2sin⁡x dx\int_{ }^{ }x^2\sin x\ dx
  • ∫e3xcos⁡(5x)dx\int_{ }^{ }e^{3x}\cos\left(5x\right)dx
  • ∫ln⁡x dx\int_{ }^{ }\ln x\ dx

Strategy


Practice: I by P

Compute ∫x2sin⁡x  ⁣dx{\displaystyle\int}x^2\sin x\de{x}.

Practice: I by P (with ln)

Find ∫ln⁡(x2) dx \int_{ }^{ }\ln\left(\frac{x}{2}\right)\ dx\ .

Practice: I by P

Evaluate ∫e3xcos⁡5x   ⁣dx{\displaystyle\int} e^{3x}\cos5x\ \de{x}.

Extra Practice
Integration by parts
Evaluate
∫xsin⁡x dx\int x\sin x\ dx
Integration by parts
Evaluate
∫xsin⁡x dx\int x\sin x\ dx
Integration by Parts
Evaluate
∫x2sin⁡x dx\int x^2\sin x\,dx
Use upper case CC to denote any constants.
Integration by parts
Evaluate
∫xsin⁡x dx\int x\sin x\ dx
Integration by parts
Evaluate
∫xsin⁡x dx\int x\sin x\ dx
Integration by Parts
Evaluate
∫x2sin⁡x dx\int x^2\sin x\,dx
Use upper case CC to denote any constants.
Integration by Parts
Evaluate
∫x2sin⁡x dx\int x^2\sin x\,dx
Use upper case CC to denote any constants.
Integration by Parts
Evaluate
∫x2sin⁡x dx\int x^2\sin x\,dx
Use upper case CC to denote any constants.
Integration by parts
Evaluate
∫xsin⁡x dx\int x\sin x\ dx