  Welcome to Live Chat

Welcome to LiveWebTutors Services, World's leading Academic solutions provider with Millions of Happy Students.

Call Back 24x7 Support Available

chat now

In A Hurry? Get A Callback Table of Content

# Constant factor rule in integration

Constant factor rule of integration is similar to the constant factor rule of derivative.Integral of f (x) = F (x) + c is the general way of writing help integration of a function f(x).For example: integral of x = (x^2)/2 + c. Here F (x) = (x^2) /2 and ‘c’ is the constant of integration.Now let us see what would be the integral of a function which is associated with a constant factor.

Integral of k * f(x).dx = k * integral of f(x).dx = k * F(x) + c

Proof: We know that, y = integral of dy. It implies that y = integral of (dy/dx).dx. Multiplying both sides by constant k we get,k * y = k * integral of (dy/dx).dx. Constant factor rule of derivative states that : d/dx (k * y) = k * dy/dx. Integratingequation 2 we get,

Integral of d/dx(k * y)dx = integral of (k * dy/dx).dx

k * y = integral of (k * dy/dx).dx …..(3)

from equation 1 and 3 we get,

integral of (k * dy/dx).dx = k * integral of (dy/dx).dx

Let us assume dy/dx = f (x)

integral of k * f(x).dx = k * integral of f(x).dx

Hence proved.

Consider an example: y = 4/x

Integral of y = integral of 4/x= integral of (4 * 1/x) (here 4 is a constant factor associated with the function 1/x). So this becomes 4 * integral of (1/x) which is equal to 4 * ln x + c. So Integral of a/x isaln x + c.

Constant factor rule and sum rule:

Integral of [k * (f (x) + g (x)] = k * integral of f(x) + k * integral of g(x)

Integral of [k * (f (x) - g (x)] = k * integral of f (x) - k * integral of g (x)

Constant factor rule and by parts rule:

Integral of [k * (f (x) * g (x)] = k * [integral of f (x) * g (x)]

Integral of f (x) *g (x) can be solved using by part’s rule.

Solved examples:

1. Find an integral of y = 4x^2

Y = 4 * integral of 9x^2).dx

Y = 4 * x^3/3 + c

= (4/3)x^3 + c

1. Find an integral of y = 2/3(1/x + sin x)

Y = (2/3) * integral of (1/x).dx + (2/3) * integral of (sin x).dx

= (2/3) * ln x + (2/3) * (-cos x) + c

= (2/3) * ln x – (2/3) * cos x + c

1. Find an integral of y = 5 * ln x

Y = 5 * integral of ln x + c

Use by parts rule to find an integration of ln x.

Consider ln x = 1 * ln x f (x) = 1 and g (x) = ln x

Integrating by parts we get, integral of ln x = x * ln x – x + c

Hence, Y = 5 * (ln x – x + c)

= 5*x.ln x – 5x + c’

Our Amazing Features
• On Time Delivery

There is no deadline that can stop our writers from delivering quality assignments to the students.

• Plagiarism Free Work

Get authentic and unique assignments by using our 100% plagiarism-free services.

• 24 X 7 Live Help

The experienced team of Live web tutors has got your back at all times of the day. Get connected with our customer support executives to receive instant and live solutions for your assignment problems.

• Services For All Subjects

We can build quality assignments in the subjects you're passionate about. Be it Engineering and Literature or Law and Marketing we have an expert writer for all.

• Best Price Guarantee

Get premium service at a pocket-friendly rate. At live web tutors, we understand the tight budget of students and thus offer our services at highly affordable prices.

FREE RESOURCES
FREE SAMPLE FILE
live review Our Mission Client Satisfaction
• I was allotted an assignment on big data which troubled me a lot. A friend recommended me to this platform and because of their experts, I was able to attain good grades with this writing task. I am thoroughly satisfied with their support.

26 Nov 2020

Alisha

• The assignment outcome has helped me thoroughly. Not only was I able to complete my assessment on time but I also understood the psychology concept covered in the paper. The solution was easy to understand and precise. Thank you for making the process easier for me.

26 Nov 2020

Mary

• I believe I could not have found a better team than them to help me with my assignments. Very talented indeed!

25 Nov 2020

Becky

• Trust me for every time I say that the team is certainly the best I could have chosen. I swear the same for you too.

25 Nov 2020

Abbie

• I am blessed to have the team beside me helping me in every step of my life. The team is really helpful at every step, and I couldn’t have been happier.

25 Nov 2020

Henry

View All Review