Tables of Integrals

You’ve learned several integration methods, but there are additional tools that can help you tackle more complex integrals or verify your answers. One of the most useful resources is integration tables.

Integration tables can be incredibly helpful, but you need to use them wisely. They’re great for:

• Quick evaluation of integrals that match standard forms
• Checking your work after solving an integral manually
• Finding patterns that might help with similar problems

Keep in mind that two completely correct solutions can look very different, so don’t panic if your answer doesn’t match the table exactly—they might still be equivalent. Here’s a perfect example of how the same integral can have multiple correct forms:

• Using trigonometric substitution with [latex]x=\tan\theta[/latex]:

  [latex]\displaystyle\int \frac{dx}{\sqrt{1+{x}^{2}}}=\text{ln}\left(x+\sqrt{{x}^{2}+1}\right)+C[/latex]

• Using hyperbolic substitution with [latex]x=\text{sinh}\theta[/latex]:

  [latex]\displaystyle\int \frac{dx}{\sqrt{1+{x}^{2}}}={\text{sinh}}^{-1}x+C[/latex]

Two antiderivatives are equivalent if their difference is just a constant. This makes sense because the derivative of a constant is zero, so [latex]\frac{d}{dx}[F(x) + C_1] = \frac{d}{dx}[G(x) + C_2][/latex] when [latex]F(x) - G(x) = \text{constant}[/latex].