Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Logarithmic functions - integration - basic.
Test Yourself 1.


 

Integrate each of the following indefinite integrals:

1. 2.
3. 4.
5. Find the primitive of . 6. Find the primitive of .
7. . 8. Integrate .

9. Integrate

10.

11.

12.
13. 14.
15. 16. .
17. (i) Show that

(ii) Hence find .

 18.
19. 20.

Evaluate each of the following definite integrals:
21. .
Answer.e2 - 1 + ln 3.		 22.
Answer.2 - ln 3.
23. 24.
25. can be written in the form

(i) Find the values of A and B.

Hint.Begin by creating a 4x using the denominator - so multiply by 2.
Then consider what you need in the second term to recreate the original expression in the numerator.

(ii) Hence find .

Answer.(i) a = 2 and b = -5.
(ii) 4 - 5ln 5.

26. (i) if the velocity (v) of a particle is described by the equation , (where t is the time during which the particle has been moving), what is the value of v when t = 0?

(ii) Show that the velocity (v) can never equal 0 for t ≥ 0.

(iii) Show that the velocity (v) can never equal 0 (i.e. it is never stationary).

(iv) Evaluating gives an expression for the distance the particle moves from t = 0
to t = 3. Determine how far the particle moved during the first 3 seconds (to the nearest cm).


Answer.(i) 1 m/sec
(ii) 6.23 m.
27. (i) Sketch the graph of .

(ii) Show that .

(iii) Hence use the Trapezoidal Rule with three function values to find an approximation to ln 3 (to 4 significant figures).

(iv) Repeat the above step using 5 function values.

(v) Explain the differences between your two values and the exact value of ln 3.

Answer.(ii) 1.367.
(iii) 1.117.
(iv) ln 3 = 1.099.		 28.