Evaluation of Definite Integrals using Substitution method Video Lecture | Mathematics (Maths) Class 12 - JEE

we want to evaluate the given definite
role first step we want to recognize
which an equation for me will use in
order to integrate and if it's helpful
we can rewrite this as the integral of 1
divided by the quantity 3x + 4
integrated from 0 to 2 with respect to X
so notice how this integral fits the
form of the integral of 1 over u
integrated with respect to U but because
our denominator is not just X here when
integrate with respect to X what the
perform u substitution or use a change
of variables so we'll let our
denominator of 3x + 4 be equal to u so
if u is equal to 3x + 4 the next step is
to find differential u differential U is
equal to the derivative of 3x plus 4
times DX which would be 3 DX notice how
we don't have 3 DX in our integral so
we'll solve this for DX by dividing both
sides by 3 notice how we have 1/3 D u
equals DX now we're going to rewrite
this in terms of U but because he flip
in some integration or for X not u for
the next step we'll leave the limits of
integration off so we'd have the
integral of again 3x plus 4 is equal to
U so we have 1 over u DX is equal to 1/3
D U so let's factor out the 1/3 and then
we have D u now it fits the integration
formula perfectly so we'll find the
antiderivative which would be 1/3 times
natural log absolute bioview or in our
case natural log absolute value of 3x +
4 now that we have our antiderivative in
terms of X we'll include our limits of
integration again which are from 0 to 2
so we'd have 1/3 times the quantity when
X is 2 we'd have natural log absolute
value of 6 plus 4 or 10 the absolute
value of 10 is 10 so we just have
natural log 10 - when x is 0 we have
natural log absolute value of 4 the
absolute value of 4 is 4 so we have
natural
for notice here we can combine these two
logs using our difference property of
logarithms we can write this as
one-third natural log of 10 divided by
four of course that simplifies the five
halves so the exact value is one-third
natural log of five halves
let's also get a decimal approximation
you have one-third natural log 5 halves
which be approximately zero point three
zero five four so we have both the exact
value and the decimal approximation a
couple more things I do want to mention
here our integrand function f of x
equals one divided by the quantity 3x +
4 is non-negative on the closed interval
from 0 to 2 which is graphed here and
therefore this def integral does give us
the area under this curve and above the
x axis on this closed interval and then
finally because we have a def integral
we can check our answer on the graphing
calculator so let's go ahead knew that
as well
so we press math press the down arrow so
I think it's option 9 function
integration press Enter we first enter
the limits of integration which are from
0 to 2 so 0 right arrow to right arrow
and now we enter the integrand function
which is 1 divided by in parentheses 3 x
+ 4
close parentheses right arrow we're
integrating with respect to X so we
enter X and enter which does verify our
work is correct I hope you found this
helpful