Select Page

Indefinite Integrals

In Physics, we know that the rate of change of displacement is velocity.

Now, if we are given the velocity of an object at any time i.e. any given instant, can we determine the displacement of that object at that instant.

Similarly, the rate of change of velocity is acceleration. If acceleration at any given time is known to us, can we identify its velocity?

Differentiation helps us to find the rate of change of any quantity with respect to another. Similarly, we need to reverse the process to find the answers to the above questions.

The answers to these questions can be given by what is called Integration.

Figure 1: https://commons.wikimedia.org/wiki/File:Terminal_Velocity.png

 

Here, we introduce the concept of Integration.

  • Integration is the inverse process of differentiation.
  • In Differentiation, we are given a function f(x) & we find its derivative i.e. f ‘ (x).
  • In Integration, we are given the derivative f ‘  (xof a function f(x). We have to find the original function i.ef(x) or the anti derivative.

This process is called anti or Integration.

 

Indefinite Integral

Let F(x)  be function which is differentiable in an interval, I. f(x) is a function such

that \frac{d}{dx} (F(x))= F ‘(x)=f(x), x \in  I Then we have,

\int {}f(x) dx = F(x) where,\int {}f(x)dx denotes the family or class of anti derivatives which is read as Indefinite integral of f with respect to x.

Note: C is a constant called the Constant of Integration.

 

For every distinct value of C, we get a different member of the family. Hence, it is called Indefinite (which is not fixed/certain) integral.

Properties of Indefinite Integrals

  1.  \int {}f ‘ (x)dx= f(x)+C
  2. \int {}[f(x)+g(x)]dx = \int {}f(x)dx + \int {} g(x)dx
  3. \int {}[f(x)-g(x)]dx = \int {}f(x)dx – \int {} g(x)dx
  4. \int {}kf(x)dx = k\int {} f(x)dx + C, where k is any non zero real number.
  5. .\int {}[k_{{1}} f(x) + k_{{2}}g(x)]dx = k_{{1}}\int {} f(x)dx + k_{{2}} \int {} g(x)dx .

 

Examples

Now let’s consider some examples on indefinite integrals.

 

Example 1Find the anti-derivative i.e. Integral of f(x) = x^8 + 3x with respect to x.

\int {}(x^8 + 3x)dx = \int {}(x^8)dx + 3\int {}(x^1)dx

\frac{ x^{{8+1}}}{8+1}+3\left ( \frac{ x^{{1+1}}}{1+1} \right )+C = \frac{x^9}{{9}} + 3\left ( \frac{x^2}{{2}} \right )+C

where C is the constant of Integration.

 

Example 2: Evaluate:\int {}(7Sin x – 3Cos x)dx

\int {}(7Sin x-3Cos x)dx = 7\int {}Sin xdx – 3\int {}Cos xdx

= 7(-Cos x) – 3(Sin x) +C = -7Cos x – 3Sin x+C

 

Example 3: Evaluate: \int {}x^2\left (1- \frac{1}{x^3} \right )dx .

\int {} x^2\left (1- \frac{1}{x^3} \right )dx = \int {}\left (x^2- \frac{1}{x} \right )dx

\int {} x^2dx – \int {}\frac{1}{x} dx = \frac{ x^{{2+1}}}{2+1} – log|x|+ C = \frac{x^3}{3} – log|x|+C

 

Example 4:  Evaluate: \int {}\frac{3+4Cos x}{Sin^2x}dx

\int {} \frac{3+4Cos x}{Sin^2x}dx = \int {}\left ( \frac{3}{Sin^2x}+ 4\frac{Cos x}{Sin^2 x} \right ) dx

\int {}(3Cosec^2 x + 4CosecxCotx)dx =3\int {}Cosec^2 xdx+4 \int {}CosecxCotxdx . 

-3Cotx -4Cosecx + C

 

Example 5Evaluate: \int {}(3x^2 -2Sin x+7\sqrt{x})dx

\int {}(3x^2 -2Sin x+7\sqrt{x})dx

=3\int {}x^2 dx -2\int {}Sin xdx+7 \int {} x^{1/2}dx=3\frac{ x^{{2+1}}}{2+1} -2(-Cosx) +7\frac{ x^{{\frac{1}{2}+1}}}{\frac{1}{2}+1}+C

= x^{3}+2Cosx +\frac{14}{3} x^{\frac{3}{2}}+C= x^{3} + 2Cosx+ \frac{14}{3}x\sqrt{x}+C

 

Check Point

Evaluate the following Integrals –

  1. \int {}(15 x^{14}+20)dx
  2. \int {}(7Cosx – 3 Sec^{2}x )dx
  3. \int {}(7 e^{x} + 34 x^{16} +17)dx
  4. \int {}(x-1)(x-2)dx
  5. \int {}\left ( \sqrt{x}-\frac{1}{\sqrt{x}} \right )dx
Answer Key
  1. \int {}(15 x^{14}+20)dx=  x^{15} + 20x +C 
  2. \int {}(7Cosx – 3 Sec^{2}x )dx = 7Sin x – 3Tan x+C
  3. \int {}(7 e^{x} + 34 x^{16} +17)dx = 7 e^{x}+2 x^{17}+17x+C
  4. \int {}(x-1)(x-2)dx= \frac{x^3}{3} – \frac{3}{2}x^2 + 2x + C
  5. \int {}\left ( \sqrt{x}-\frac{1}{\sqrt{x}} \right )dx  =  \frac{3}{2} x^{3/2} – 2 x^{1/2} +C .

Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. eTutorWorld provides the best K-12 Online Tutoring Services so you can learn from the comfort and safety of your home at an affordable cost.

Be it an exam, class test or a quiz, eTutorWorld’s Math, Science and English tutors are responsible for your academic progress. Meet your personal coach at your convenient day and time to get help for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. All our tutors are graduates and bring with them years of teaching experience to the tutoring lessons.

Our ‘Learning by Design’ methodology makes sure that each student is at the center of the teaching-learning process. All tutoring sessions start with a question to the student ‘What do you want to learn today?’ Hence, tutors diagnose your skills and recognize your requirements before the actual tutoring happens. Post every tutoring session, an individualized worksheet is emailed to the student to assimilate learned concepts. Regular formative assessments are used to evaluate a student’s understanding of the subject.

The state of the art technology used is stable, user friendly and safe. All you need is a computer or a tablet and an internet connection. The easy-to-use web conferencing software requires a one time download, using which the student can talk and chat to the tutor, annotate on an interactive shared whiteboard or even share documents, assignments or worksheets.

All tutoring sessions are recorded and made available for a month so you can review concepts taught.

Email or call our support team with any issues or questions – we are here for you 24X7.

Also, download free printable math and science worksheets in pdf format and solve SCAT and SSAT Practice Tests online. Sign up for a Free Trial Lesson Today!

Thousands have taken the eTutorWorld Advantage – what are you waiting for?

© 2019 eTutorWorld - Online Tutoring and Test Prep | All rights reserved