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27.3 Examples for Drill

We now present a list of integrals for you to examine. You should look at them all and try to decide how to attack each. Once you have an idea how to do them, you should do a few of each kind. You will not do integrals fluently until you have done a dozen or so, and have fallen into all the standard traps, and then you will know what to avoid.

Here are some hints which probably will not help until you make these mistakes yourself.

The most common mistake is forgetting a small factor when changing variables and relating dx with some du.

Losing a small factor when copying one line to the next is also very common as is dropping a sign.

Forgetting that you are integrating and writing a derivative instead of an integral of some common function which you integrate by inspection where the integral is called for, is fairly common.

Failing to adjust the limits of integration upon changing variables is another common error.

Failing to pay attention to the singularities of the integrand, and not noticing that you have integrated over one, is rarer but only because you rarely run into the problem.

Carelessly integrating ( f ( x ) ) k d x to get f k + 1 k + 1 without the presence of any f ' factor.

Exercises: Describe what you would do to evaluate each of the following integrals. Then do five of them.

1. ∫ ( 2 x + 4 ) − 1 / 3 d x

2. ∫ d x e x sin ⁡ x

3. ∫ ( 4 x 2 − 11 ) − 1 / 3 x d x

4. ∫ d x x e 3 x cos ⁡ 2 x .

5. ∫ d x x + 1

6. ∫ d x ( x + 1 ) 2

7. ∫ d x 3 x 2 + 2 x − 1

8. ∫ d x 3 x 2 + 2 x + 1

9. ∫ 2 x d x ( x − 3 ) ( x + 1 )

10. ∫ x 3 d x ( x − 3 ) ( x + 1 ) 2

11. ∫ d x ( x + 5 ) 3 x 2 + 2 x + 1

12. ∫ ( sin ⁡ x ) ( sec ⁡ x ) 3 d x

13. ∫ e 3 x ( sin ⁡ x ) 2 d x

14. ∫ ( sin ⁡ x ) 2 ( cos ⁡ x ) 4 d x

15. ∫ e − x 2 x d x

16. ∫ x cos ⁡ x d x

17. ∫ x 2 ( ln ⁡ x ) d x

18. ∫ x ( log ⁡ 10 x ) d x

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