Calculus I - Computing Indefinite Integrals

Introduction

• Focus on computing indefinite integrals.
• Basic indefinite integrals discussed include:
  • Powers of (x)
  • Constants
  • Trigonometric functions
  • Exponential and logarithm functions
  • Inverse trigonometric and hyperbolic functions

Basic Indefinite Integrals

Power of (x)

• [\int{{{x^n},dx}} = \frac{{{x^{n + 1}}}}{{n + 1}} + c,,,,,n \ne - 1]
• Add one to the exponent, divide by the new exponent.
• Avoid (n = - 1) to prevent division by zero.

Constants

• [\int{{k,dx}} = kx + c]
• Simple integration as differentiation of a constant term.

Trigonometric Functions

• [\begin{array}{ll} \int{{\sin x,dx}} = - \cos x + c & \int{{\cos x,dx}} = \sin x + c\ \int{{{{\sec }^2}x,dx}} = \tan x + c & \int{{\sec x\tan x,dx}} = \sec x + c\ \int{{{{\csc }^2}x,dx}} = - \cot x + c & \int{{\csc x\cot x,dx}} = - \csc x + c \end{array}]
• Pay attention to the signs to avoid confusion with derivatives.
• Other trig functions require the Substitution Rule.

Exponential and Logarithmic Functions

• [\int{{{{\bf{e}}^x},dx}} = {{\bf{e}}^x} + c]
• [\int{{{a^x},dx}} = \frac{{{a^x}}}{{\ln a}} + c]
• [\int{{\frac{1}{x},dx}} = \ln \left| x \right| + c]
• Logarithmic integration taught in Calculus II.

Inverse Trigonometric and Hyperbolic Functions

• [\begin{array}{ll} \int{{\frac{1}{{{x^2} + 1}},dx}} = {\tan ^{ - 1}}x + c & \int{{\frac{1}{{\sqrt {1 - {x^2}} }},dx}} = {\sin ^{ - 1}}x + c\ \int{{\sinh x,dx}} = \cosh x + c & \int{{\cosh x,dx}} = \sinh x + c \end{array}]
• [\int{{\frac{1}{{\sqrt {1 - {x^2}} }},dx}} = - {\cos ^{ - 1}}x + c]
• Different forms exist for inverse trig integrals; first form is standard.

Examples of Indefinite Integrals

Example 1

1. (\int{{5{t^3} - 10{t^{ - 6}} + 4,dt}})

   • Use formulas: (\frac{5}{4}{t^4} + 2{t^{ - 5}} + 4t + c)

2. (\int{{{x^8} + {x^{ - 8}},dx}})

   • (\frac{1}{9}{x^9} - \frac{1}{7}{x^{ - 7}} + c)

3. (\int{{3\sqrt[4]{{{x^3}}} + \frac{7}{{{x^5}}} + \frac{1}{{6\sqrt x}},dx}})

   • Rewrite expressions: (\frac{12}{7}{x^{\frac{7}{4}}} - \frac{7}{4}{x^{ - 4}} + \frac{1}{3}{x^{\frac{1}{2}}} + c)

4. (\int{{dy}})

   • (y + c)

5. (\int{{\left( {w + \sqrt[3]{w}} \right)\left( {4 - {w^2}} \right)dw}})

   • Expand and integrate: (2{w^2} - \frac{1}{4}{w^4} + 3{w^{\frac{4}{3}}} - \frac{3}{10}{w^{\frac{10}{3}}} + c)

6. (\int{{\frac{{4{x^{10}} - 2{x^4} + 15{x^2}}}{{{x^3}}},dx}})

   • Simplify and integrate: (\frac{1}{2}{x^8} - {x^2} + 15\ln \left| x \right| + c)

Example 2

1. (\int{{3{{\bf{e}}^x} + 5\cos x - 10{{\sec }^2}x,dx}})

   • (3{{\bf{e}}^x} + 5\sin x - 10\tan x + c)

2. (\int{{2\sec w\tan w + \frac{1}{{6w}},dw}})

   • Simplify: (2\sec w + \frac{1}{6}\ln \left| w \right| + c)

3. (\int{{\frac{{23}}{{{y^2} + 1}} + 6\csc y\cot y + \frac{9}{y},dy}})

   • (23{\tan ^{ - 1}}y - 6\csc y + 9\ln \left| y \right| + c)

4. (\int{{\frac{3}{{\sqrt {1 - {x^2}} }} + 6\sin x + 10\sinh x,dx}})

   • (3{\sin ^{ - 1}}x - 6\cos x + 10\cosh x + c)

5. (\int{{\frac{{7 - 6{{\sin }^2}\theta }}{{{{\sin }^2}\theta }},d\theta }})

   • Simplify: (- 7\cot \theta - 6\theta + c)

Example 3

• (\int{{\sin \left( {\frac{t}{2}} \right)\cos \left( {\frac{t}{2}} \right),dt}})
  • Use double angle formula: (- \frac{1}{2}\cos \left( t \right) + c)

Example 4

• (f\left( x \right)) from (f'\left( x \right)) and initial conditions.

  • (x = 0), (c = 10): (f\left( x \right) = {x^4} - 9x - 2\cos x + 7{{\bf{e}}^x} + 10)

• (f\left( x \right)) from (f''\left( x \right)) with two conditions:

  • Solve system for (c) and (d).
  • (f\left( x \right) = 4{x^{\frac{5}{2}}} + \frac{1}{4}{x^5} + 3{x^2} - \frac{13}{2}x - 2)

Conclusion

• Introduction to integration started.
• Understand the process and rules of indefinite integrals.
• The integration represents finding the function that was differentiated to achieve the integrand.