List of Notes - Category: Integration Techniques
Substitution Rule is in fact one of the most powerful rules for integration.
Suppose we want to find `int f(x)dx`.
If we can find such functions `g` and `v` that `f(x)dx=g(v(x))v'(x)dx` then according to Chain Rule `d/(dx)G(v(x))=G'(v(x))v'(x)=g(v(x))v'(x)` where `G'=g`.
It is easy to compute integral `int e^xdx` but how to handle integrals like `int xe^xdx`? In general if you have under integral product of functions that can be easily integrated separately then you should use integration by parts.
Integration by Parts and Substitution Rule will not help if we directly apply them to integrals like `int cos^5(x)dx`. Because if `u=cos(x)` then `du=sin(x)dx`. In order to integrate powers of cosine, we would need an extra `sin(x)` factor. Similarly, a power of sine would require an extra `cos(x)` factor. Thus, here we separate one cosine factor and convert the remaining factor to the expression involving sine.
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.
Recall that trignomeric identity states `cos^2(x)+sin^2(x)=1`.
Consider integral `int (4x-6)/(x^2-3x+2)dx`.
This integral can be easily evaluated using substitution `u=x^2-3x+2`.
Indeed, `du=(2x-3)dx`. So, integral becoms `2 int (du)/u=2ln|u|+C=2ln|x^2-3x+2|+C`.
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.
No need to say that integration is a difficult task. Following tips will help you in integrating.
- You should have good knowledge of Calculus I. Without this, studying Calculus II is very difficult.
- You should memorize integrals of basic functions. Integral `int sec(x)tan(x)dx` looks difficult unless you know that `int sec(x)tan(x)dx=sec(x)+C`.
- 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(x)tan^2(x)dx` . In fact we can if we set `u=tan(x)` and `dv=sec(x)tan(x)dx` then `du=sec^2(x)dx` and `v=sec(x)`.
- Simplify integrand as much as possible to integrate it. For example, `int sin^2(x)dx` can't be integrated directly, however we can simplify it by using double angle formula: `int sin^2(x)dx=1/2 int (1-cos(2x))dx=1/2(x-1/2 sin(2x))+C`. Sometimes just rewriting tricks needed. For example, multiplying numerator and denominator by same value: `int sec(x)dx=int (sec(x)(sec(x))+tan(x))/(sec(x)+tan(x))dx=int (sec^2(x)+sec(x)tan(x))/(tan(x)+sec(x))dx=` `=ln|tan(x)+sec(x)|+C`
- 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.
- Check whether "simple" substitution will work. Integral `int xsqrt(x^2+1)dx` can be solved using trigonometric substitution, but there is simpler substitution that will also work, namely, `u=x^2+1` .
- 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.
- Try to relate integral we already know how to do or to integrals listed in Table of Integrals. Typical example is `int cos(x)sqrt(sin^2(x)+1)dx` . Using substitution `u=sin(x)` this integral is converted to `int sqrt(u^2+1)du` which we already know how to do from Trigonometric Substitutions Note (Substitution `t=tan(u)` ). Note that we use substitution rule twice.
- 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.
- 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.
- Not all integrals can be expressed in terms of elementary fuctions. For, example we can't find `int e^(-x^2)dx` , `int (sin(x))/xdx` etc. Just remember about it.
Now, let's work out a couple of examples.