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

Calculus - Integration by substitution - basic functions with same powers of x.
Test Yourself 1.


 

The process for this extension technique is very similar to that used for the Reverse Chain Rule.
The difference is that the Reverse Chain Rule uses only derivatives.
This technique uses a derivative and the rearranged original substitution function .

Indefinite integrals. 1. Find using the substitution u = 1 - x.
  2. Using the substitution u = 2 + x, find .
  3. By using the substitution u = x + 1 or otherwise, find

  4.
  5. Find using the substitution m = t2 + 1.
  6.
Definite integrals. 7. Evaluate .
Answer.-256/5.
  8. By using the substitution u = 4x + 1, evaluate

.

Answer.1/4(ln 5 - 0.8).
  9. Evaluate by using the substitution
u = 2t - 1.
  10.
  11. Find a primitive of .
  12. Find the anti-derivative of .
  10. Find a primitive of .
  11. Find the anti-derivative of .
  12. The curve is graphed below.

For this curve, evaluate .
 

13. Evaluate using the substitution u = 2x + 1.

Answer.4 - ln 3.
  14. Show that .
Trigonometric functions. 15. Find the primitive of .
  16. Find the anti-derivative of using the substitution u = sin x.
  17. Find the anti-derivative of using the substitution u = sin x.
  18. Find the primitive of .
  19. Evaluate .
Answer.-1/4.
  20. Show that .
  21.