Integral Calculus- Recap

Integral Calculus- Recap

Integration as the reverse of Differentiation

If differentiation can calculate the infinitesimal of a given quantity, then integration can add up these infinitesimal quantities.

For example, if the differentiation of xⁿ is nx^n-1 then the integration of nx^n-1 is xⁿ

^{With this result in mind, we will take a look at basic integration results.}

Basic Integration results

Algebraic results

Algebraic results

•  $\int_{ }^{ }x^{n\ }dx\ =\ \frac{x^{n+1}}{n\ +\ 1}\ +\ C$  

•  $\int_{ }^{ }\frac{1}{x}\ dx\ =\ \ln\left|x\right|\ +\ C$  

•  $\int_{ }^{ }a^{x\ }dx\ =\ \frac{a^x}{\ln a}\ +\ C$  

•  $\int_{ }^{ }e^{x\ }dx\ =\ e^x\ +\ C$  

Trigonometric Results (Differentiation)

•  $\frac{d\left(\sin x\right)}{dx}\ =\ \cos x$  

•  $\frac{d\left(\cos x\right)}{dx}\ =\ -\sin x$  

•  $\frac{d\left(\tan x\right)}{dx}\ =\ \sec^2x$  

•  $\frac{d\left(\sec x\right)}{dx}\ =\ \sec x\ \tan x$  

•  $\frac{d\left(\operatorname{cosec}x\right)}{dx}\ =\ -\operatorname{cosec}x\ \cot x$  

•  $\frac{d\left(\cot x\right)}{dx}\ =\ -\operatorname{cosec}^2x$  

Trigonometric Results (Integration)

•  $\ \int_{ }^{ }\cos x\ =\ \sin x\ +\ C$  

•  $\ \int_{ }^{ }-\sin x\ =\ \cos x\ +\ C$  

•  $\ \int_{ }^{ }\sec^2x\ =\ \tan x\ +\ C$  

•  $\ \int_{ }^{ }\sec x\ \tan x\ =\ \sec x\ +\ C$  

•  $\ \int_{ }^{ }-\operatorname{cosec}x\ \cot x\ =\ \operatorname{cosec}x\ +\ C$  

•  $\ \int_{ }^{ }-\operatorname{cosec}^2x\ =\ \cot x\ +\ C$  

Let's get to a bit higher level stuff

Advanced Trigonometric integrals

Advanced Trigonometric Integrals

•  $\int_{ }^{ }\sec x\ dx\ =\ \ln\left|\tan x\ +\ \sec x\right|\ +\ C$  

•  $\int_{ }^{ }\operatorname{cosec}x\ dx\ =-\ln\left|\cot x\ +\ \operatorname{cosec}x\right|\ +\ C\ =\ \ln\left|\tan\left(\frac{x}{2}\right)\right|\ +\ C$  

•  $\int_{ }^{ }\tan x\ dx\ =\ \ln\left|\sec x\right|\ +\ C$  

•  $\int_{ }^{ }\cot x\ dx\ =\ \ln\left|\sin x\right|\ +\ C$  

Advanced Algebraic Integrals  (1/4)

•  $\int\ \frac{dx}{\sqrt{a^2\ -\ x^2}}\ =\ \sin^{-1}\left(\frac{x}{a}\right)\ +\ C$  

•  $\int\ \frac{dx}{x\sqrt{x^2\ -\ a^2}}\ =\ \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)+\ C$  

•  $\int\ \frac{dx}{x^2\ +\ a^2}\ =\frac{1}{a}\ \tan^{-1\ }\left(\frac{x}{a}\right)\ +\ C$  

Advanced Algebraic Integrals (2/4)

•  $\int\ \frac{dx}{x^{2\ }-\ a^2}\ =\ \frac{1}{2a}\ \ln\left|\frac{x\ -a}{x\ +\ a}\right|\ +\ C$  

•  $\int\ \frac{dx}{-x^{2\ }+\ a^2}\ =\ \frac{1}{2a}\ \ln\left|\frac{x\ +a}{-x\ +\ a}\right|\ +\ C$  

• Notice how the numerator's 'a' sign depends and denominator's 'x' sign depends

Advanced Algebraic Integrals (3/4)

•  $\int\ \frac{dx}{\sqrt{x^{2\ }\pm\ a^2}}\ =\ \ \ln\left|x\ +\sqrt{x^{2\ }\pm\ a^2}\ \right|\ +\ C$ 

Advanced Algebraic Integrals (4/4)

•  $\ \int_{ }^{ }\sqrt{x^{2\ }\pm\ a^2}\ dx\ =\ \frac{x}{2}\sqrt{x^{2\ }\pm\ a^2}\ \ \pm\ \frac{a^2}{2}\ln\left|x\ +\ \sqrt{x^{2\ }\pm\ a^2}\right|\ +\ C$  

•  $\int_{ }^{ }\sqrt{-x^{2\ }+\ a^2}\ dx\ =\ \frac{x}{2}\sqrt{-x^{2\ }+\ a^2}\ \ +\ \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right)+\ C$  

• Notice how the function behaves differently when the coefficient of x is negative.

• The signs of each term are heavily dependant on the Integrand.

Few properties of Integration

Lets learn about basics of evaluating integrals

Properties

•  $\int\left[f\left(x\right)\ \pm\ g\left(x\right)\right]dx\ =\ \int f\left(x\right)dx\ \pm\ \int g\left(x\right)dx$  

•  $\int kf\left(x\right)dx\ =\ k\int f\left(x\right)dx$  

•  $\frac{\text{d}\int f\left(x\right)\ dx}{\text{d}x}\ =\ f\left(x\right)$  

•  $\int f\left(ax\ +\ b\right)dx\ =\ \frac{1}{a}f\left(ax\ +\ b\right)\ +\ C$  

Integrating a function w.r.t another function

Integrate w.r.t another function

 $\int x^2\ d\left(3x^3\right)\ =\ \int\ x^2\ \ \left(9x^2\right)dx\$  

 $=\ 9\int\ x^4\ dx\$  

 $=\ 9\left(\frac{x^5}{5}\ \right)\ +\ C$  

Multiple Choice

 $\int2^xe^xdx\ is?$  

•  $\frac{\left(2e\right)^x}{1\ +\ \ln2}\ +\ C$  

•  $\frac{\left(2e\right)^x}{1\ -\ \ln2}\ +\ C$  

•  $\left(2e\right)^x\ +\ C$  

•  $2e^x\ +\ C$  

Now is a good time to try writing down all the formulas you have learned without seeing

If you are done, you can proceed to the next section

Using trigonometric Identities in Integration trigonometric functions

Will also be useful in substitution

Now is a good time to try writing down all the formulas you have learned without seeing

If you are done, you can proceed to the next section

Algebraic Identities

Here is a relatively easier section for warm up!

Cube formulas

•  $\left(a\ \pm\ b\right)^3\ =\ a^3\ \pm\ b^{3\ }\pm\ 3ab\left(a\ \pm b\right)$  

•  $a^{3\ }\ -\ \ b^3\ \ =\ \left(a\ -\ b\right)\left(a^{2\ }+\ b^{2\ }+\ ab\right)$  

•  $a^{3\ }\ +\ \ b^3\ \ =\ \left(a\ +\ b\right)\left(a^{2\ }+\ b^{2\ }-\ ab\right)$  

• Carefully observe the signs in the differences of cubes identity.

Multiple Choice

 $Evaluate\ \int\ \frac{\left(\sqrt{x}\ +\ 1\right)\ \left(x^{2\ }\ +\ \sqrt{x}\right)}{x\sqrt{x}\ +\ x\ +\ \sqrt{x}}dx\ \ \left(Use\ clever\ factorization\right)$  

•  $\frac{x^2}{2}\ +\ x\ +\ C$  

•  $\frac{x^2}{2}\ -\ x\ +\ C$  

•  $\frac{x^{ }}{2}\ \ +\ C$  

•  $x\sqrt{x}\ +\ x\ +\ c$  

Solution  $\int\ \frac{\left(\sqrt{x}\ +\ 1\right)\ \left(x^{2\ }\ +\ \sqrt{x}\right)}{x\sqrt{x}\ +\ x\ +\ \sqrt{x}}dx$  

 $=\int\ \frac{\left(\sqrt{x}\ +\ 1\right)\ \sqrt{x}\left(\left(\sqrt{x}\right)^{3\ }\ +\ 1^3\right)}{x\sqrt{x}\ +\ x\ +\ \sqrt{x}}dx\ \ \left(Numerator\ factorization\right)$  
 $=\int\ \frac{\left(\sqrt{x}\ +\ 1\right)\ \sqrt{x}\left(\left(\sqrt{x}\right)^{3\ }\ +\ 1^3\right)}{\sqrt{x}\left(x\ +\ \sqrt{x}\ +\ 1\right)}dx\ \ \left(Denomiator\ factorization\right)$  

Solution continuation

 $=\int\ \frac{\left(\sqrt{x}\ +\ 1\right)\left(\sqrt{x}\ -\ 1\right)\ \ \left(x\ +\ \sqrt{x}\ +\ 1\right)}{\left(x\ +\ \sqrt{x}\ +\ 1\right)}dx$                             (Using a3 + b3 algebraic identity and canceling out root(x) terms)
 $=\int\ \left(\sqrt{x}\ +\ 1\right)\left(\sqrt{x}\ -\ 1\right)\ dx$  

 $=\int\ \left(x^{\ }-\ 1\right)\ dx\ =\ \frac{x^2}{2}\ -\ x\ +\ C$  

Revise your trigonometric formulas to attend the next two questions

If you are confident enough, proceed.

Multiple Choice

 $Evaluate\ \int\sin x\ d\left(\cos x\right)\$  

•  $\frac{\sin2x}{4}\ -\ \frac{x}{2}\ +\ C$  

•  $\frac{\sin x}{4}+\ \ \frac{x}{2}\ +\ C$  

•  $\frac{\sin2x}{2}\ -\ \frac{x}{2}\ +\ C$  

•  $\frac{\cos2x}{2}\ +\ \frac{x}{2}\ +\ C$  

Multiple Choice

 $Evaluate\ \int\ \frac{\left(\sqrt{x}\ +\ 1\right)\ \left(x^{2\ }\ +\ \sqrt{x}\right)}{x\sqrt{x}\ +\ x\ +\ \sqrt{x}}dx\ \ \left(Use\ clever\ factorization\right)$  

•  $\frac{x^2}{2}\ +\ x\ +\ C$  

•  $\frac{x^2}{2}\ -\ x\ +\ C$  

•  $\frac{x^{ }}{2}\ \ +\ C$  

•  $x\sqrt{x}\ +\ x\ +\ c$  

Try these sums as practice

•  $\int\tan x\ \tan2x\ \tan\ 3x\ dx\$  

•  $\int\ \frac{\left(x^{3\ }+\ 8\right)\ \left(x\ -\ 1\right)}{x^{2\ }-2x\ +\ 4}dx$  

Trigonometry sum to product and product to sums identity