Separation of Variables

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Calculus I

Derivatives

Differentiation Rules

General Formulas

Exponential and Logarithmic Formulas

Trigonometric Formulas

Inverse Trigonometric Formulas

Hyperbolic Formulas

Inverse Hyperbolic Formulas

Antiderivatives

Antidifferentiation Rules

Basic Forms

Integration Strategy

This is a suggested strategy to use while attempting to evaluating an integral. This can be used for indefinite and definite integrals.

Fundamental Theorem of Calculus - Part I

Use known formulas from your knowledge of derivatives to find antiderivatives.

Example 01: sec(x)tan(x)dx

Since we know ddxsec(x)=sec(x)tan(x), then

ddxsec(x)dx=sec(x)tan(x)dx, and

sec(x)+C=sec(x)tan(x)dx

Thus, sec(x)tan(x)dx=sec(x)+C

Example 02: e3xdx

Since we know ddxe3x=3e3x, then

ddxe3xdx=3e3xdx, which is equivalent to

ddxe3xdx=3e3xdx, and

13ddxe3xdx=e3xdx

Thus, e3xdx=13e3x+C

Rewrite the Integrand

Use algebra or trigonometry to rewrite the integrand.

Example 03: x1x21dx

x1x21dx=x1(x1)(x+1)dx

1x+1dx ln(x+1)+C

Example 04: x2x21dx

x2x21dxd=x2+0x21dx

x21+1x21dx

(x21x21+1x21)dx

(1+1x21)dx

(111x2)dx

xarctanh(x)+C

Example 05: tan(x)sin(x)dx

tan(x)sin(x)dx=sin(x)cos(x)1sin(x)dx

1cos(x)dx

sec(x)dx

ln|sec(x)+tan(x)|+C

Example 06: sin2(x)dx

sin2(x)dx=12(1cos(2x))dx

12(x12sin(2x))+C

Additional Methods

Practice 01

Evaluate the following integrals:

  1. (3x2+x5)exdx

  2. 11x2dx

  3. x1x2dx

  4. x21x2dx

  5. x31x2dx

  6. x21+x2dx

  7. x31+x3dx

  8. 1+x21+xdx

  9. sec2(x)1+tan(x)dx

  10. 1sin(x)cos(x)dx

  11. xx23x+2dx

  12. x23x+2xdx

  13. sin1(x)dx

  14. sin3(x)cos5(x)dx

  15. sin3(x)cos2(x)dx

  16. tan3(x)sec3(x)dx

  17. tan4(x)sec4(x)dx

  18. cos4(x)dx

  19. 1xln(x)dx

  20. xln(x)dx

  21. exdx

  22. x24xdx

  23. 1x2x2+1dx

  24. e3xcos(5x)dx

  25. 1ex+exdx

  26. x+1x2+4dx

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Last Modified: Monday, 31 August 2020 12:24 EDT