Assignments

The integrals below can be calculated by using a special method: integration by parts. For some assignments different techniques have to be applied. Always verify the result.

1. Calculate:

$\displaystyle\int{\displaystyle\frac{1}{x}\ln(x)}dx$

Solution

2. Calculate:

$\displaystyle\int{e^x\sin(x)}dx$

Solution

3. Calculate:

$\displaystyle\int{\sin^2(x)}dx$

Solution

4. Calculate:

$\displaystyle\int{x\sin(x)}dx$

Solution

5. Calculate:

$\displaystyle\int{\sin[\ln(t)]}dt$

Solution

6. Calculate:

$\displaystyle\int{\ln(\sqrt{x})}dx$

Solution

7. Calculate:

$\displaystyle\int{x\ln(x)}dx$

Solution

8. Write the following integral in a recursive form:

$\displaystyle\int{[\ln(x)]^n}dx$

Solution

9. Show the following formula (we have already applied this a number of times):

$\displaystyle\int{f(x)}dx=xf(x)-\displaystyle\int{xf'(x)}dx$

Solution

10. In assignment 3 it was asked to calculate the following integral and we applied integration by parts. Is there another way and if so, which?

$\displaystyle\int{\sin^2(x)}dx$

Solution

Integration by parts

Assignments

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