Category: Integration Techniques

Substitution (Change of Variable) Rule

The substitution rule is in fact one of the most powerful rules for integration.

Suppose that we want to find $$$\int{f{{\left({x}\right)}}}{d}{x}$$$.

If we can find such functions $$${g{}}$$$ and $$${v}$$$ that $$${f{{\left({x}\right)}}}{d}{x}={g{{\left({v}{\left({x}\right)}\right)}}}{v}'{\left({x}\right)}{d}{x}$$$, according to the chain rule, $$$\frac{{d}}{{{d}{x}}}{G}{\left({v}{\left({x}\right)}\right)}={G}'{\left({v}{\left({x}\right)}\right)}{v}'{\left({x}\right)}={g{{\left({v}{\left({x}\right)}\right)}}}{v}'{\left({x}\right)}$$$, where $$${G}'={g{}}$$$.

Integration by Parts

It is easy to compute the integral $$$\int{{e}}^{{x}}{d}{x}$$$, but how to handle integrals like $$$\int{x}{{e}}^{{x}}{d}{x}$$$? In general, if you have under the integral sign a product of functions that can be easily integrated separately, you should use integration by parts.

Integrals Involving Trig Functions

Integration by Parts and Substitution Rule will not help if we directly apply them to integrals like $$$\int{{\cos}}^{{5}}{\left({x}\right)}{d}{x}$$$. Because if $$${u}={\cos{{\left({x}\right)}}}$$$ then $$${d}{u}={\sin{{\left({x}\right)}}}{d}{x}$$$. In order to integrate powers of cosine, we would need an extra $$${\sin{{\left({x}\right)}}}$$$ factor. Similarly, a power of sine would require an extra $$${\cos{{\left({x}\right)}}}$$$ factor. Thus, here we separate one cosine factor and convert the remaining factor to the expression involving sine.

Trigonometric Substitutions In Integrals

Trigonometric Substitutions are especially useful when we want to get rid of $$$\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}$$$, $$$\sqrt{{{{x}}^{{2}}+{{a}}^{{2}}}}$$$ and $$$\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}$$$ under integral sign.

Integrals Involving Rational Functions

Consider integral $$$\int\frac{{{4}{x}-{6}}}{{{{x}}^{{2}}-{3}{x}+{2}}}{d}{x}$$$.

This integral can be easily evaluated using substitution $$${u}={{x}}^{{2}}-{3}{x}+{2}$$$.

Indeed, $$${d}{u}={\left({2}{x}-{3}\right)}{d}{x}$$$. So, integral becoms $$${2}\int\frac{{{d}{u}}}{{u}}={2}{\ln}{\left|{u}\right|}+{C}={2}{\ln}{\left|{{x}}^{{2}}-{3}{x}+{2}\right|}+{C}$$$.

Integration Formulas (Table of Indefinite Integrals)

Below is a table of Indefinite Integrals. With this table and integration techniques, you will be able to find majority of integrals.

It is also worth noting that unlike derivative (we can find derivative of any function), we can't find integral of any function: this means that we can't find integral in terms of functions we know.

Integration Strategy

No need to say that integration is a difficult task. Following tips will help you in integrating.

  1. You should have good knowledge of Calculus I. Without this, studying Calculus II is very difficult.
  2. You should memorize integrals of basic functions. Integral $$$\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}$$$ looks difficult unless you know that $$$\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}={\sec{{\left({x}\right)}}}+{C}$$$.
  3. You should master all integration techniques. You can't just memorize formulas, you need to understand them and know how they work. For example, you can memorize formula for integration by parts, but it doesn't mean that you can successfully use it. You need to look at integral and realize whether you can use integration by parts and if yes then how to use it. For example, it is note very clear whether we can use integration by parts for $$$\int{\sec{{\left({x}\right)}}}{{\tan}}^{{2}}{\left({x}\right)}{d}{x}$$$. In fact we can if we set $$${u}={\tan{{\left({x}\right)}}}$$$ and $$${d}{v}={\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}$$$ then $$${d}{u}={{\sec}}^{{2}}{\left({x}\right)}{d}{x}$$$ and $$${v}={\sec{{\left({x}\right)}}}$$$.
  4. Simplify integrand as much as possible to integrate it. For example, $$$\int{{\sin}}^{{2}}{\left({x}\right)}{d}{x}$$$ can't be integrated directly, however we can simplify it by using double angle formula: $$$\int{{\sin}}^{{2}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}\int{\left({1}-{\cos{{\left({2}{x}\right)}}}\right)}{d}{x}=\frac{{1}}{{2}}{\left({x}-\frac{{1}}{{2}}{\sin{{\left({2}{x}\right)}}}\right)}+{C}$$$. Sometimes just rewriting tricks needed. For example, multiplying numerator and denominator by same value: $$$\int{\sec{{\left({x}\right)}}}{d}{x}=\int\frac{{{\sec{{\left({x}\right)}}}{\left({\sec{{\left({x}\right)}}}\right)}+{\tan{{\left({x}\right)}}}}}{{{\sec{{\left({x}\right)}}}+{\tan{{\left({x}\right)}}}}}{d}{x}=\int\frac{{{{\sec}}^{{2}}{\left({x}\right)}+{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}}}{{{\tan{{\left({x}\right)}}}+{\sec{{\left({x}\right)}}}}}{d}{x}=$$$ $$$={\ln}{\left|{\tan{{\left({x}\right)}}}+{\sec{{\left({x}\right)}}}\right|}+{C}$$$
  5. Identify the type of integral. If integrand is rational expression, then maybe it is possbile to use partial fraction decomposition. If integrand is product of polynomial, exponential, trigonometric or logartihmic functions then integration by parts may work. If integrand involves roots then trigonometric substitution may work.
  6. Check whether "simple" substitution will work. Integral $$$\int{x}\sqrt{{{{x}}^{{2}}+{1}}}{d}{x}$$$ can be solved using trigonometric substitution, but there is simpler substitution that will also work, namely, $$${u}={{x}}^{{2}}+{1}$$$.
  7. Prepare to use multiple technniques. Many of integrals require use of more than one technique. For example you use substitution rule, then integration by parts and then again substitution rule.
  8. Try to relate integral we already know how to do or to integrals listed in Table of Integrals. Typical example is $$$\int{\cos{{\left({x}\right)}}}\sqrt{{{{\sin}}^{{2}}{\left({x}\right)}+{1}}}{d}{x}$$$. Using substitution $$${u}={\sin{{\left({x}\right)}}}$$$ this integral is converted to $$$\int\sqrt{{{{u}}^{{2}}+{1}}}{d}{u}$$$ which we already know how to do from Trigonometric Substitutions Note (Substitution $$${t}={\tan{{\left({u}\right)}}}$$$ ). Note that we use substitution rule twice.
  9. Try everyhing you can. If all techniques that you've tried failed, look at integral from another side and try technique that you didn't use.
  10. Practice, practice and once more practice. You will master integration only after you will solve a whole bunch of integrals. In this case you will be able to "see" correct substitutions and identify integral.
  11. Not all integrals can be expressed in terms of elementary fuctions. For, example we can't find $$$\int{{e}}^{{-{{x}}^{{2}}}}{d}{x}$$$, $$$\int\frac{{{\sin{{\left({x}\right)}}}}}{{x}}{d}{x}$$$ etc. Just remember about it.

Now, let's work out a couple of examples.