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Suppose the rate of growth of the fly population is given by g ( t ) = e 0.01 t , and the initial fly population is 100 flies. How many flies are in the population after 15 days?

There are 116 flies.

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Evaluating a definite integral using substitution

Evaluate the definite integral using substitution: 1 2 e 1 / x x 2 d x .

This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on e as a power of x , then bring the x 2 in the denominator up to the numerator using a negative exponent. We have

1 2 e 1 / x x 2 d x = 1 2 e x −1 x −2 d x .

Let u = x −1 , the exponent on e . Then

d u = x −2 d x d u = x −2 d x .

Bringing the negative sign outside the integral sign, the problem now reads

e u d u .

Next, change the limits of integration:

u = ( 1 ) −1 = 1 u = ( 2 ) −1 = 1 2 .

Notice that now the limits begin with the larger number, meaning we must multiply by −1 and interchange the limits. Thus,

1 1 / 2 e u d u = 1 / 2 1 e u d u = e u | 1 / 2 1 = e e 1 / 2 = e e .
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Evaluate the definite integral using substitution: 1 2 1 x 3 e 4 x −2 d x .

1 2 1 x 3 e 4 x −2 d x = 1 8 [ e 4 e ]

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Integrals involving logarithmic functions

Integrating functions of the form f ( x ) = x −1 result in the absolute value of the natural log function, as shown in the following rule. Integral formulas for other logarithmic functions, such as f ( x ) = ln x and f ( x ) = log a x , are also included in the rule.

Rule: integration formulas involving logarithmic functions

The following formulas can be used to evaluate integrals involving logarithmic functions.

x −1 d x = ln | x | + C ln x d x = x ln x x + C = x ( ln x 1 ) + C log a x d x = x ln a ( ln x 1 ) + C

Finding an antiderivative involving ln x

Find the antiderivative of the function 3 x 10 .

First factor the 3 outside the integral symbol. Then use the u −1 rule. Thus,

3 x 10 d x = 3 1 x 10 d x = 3 d u u = 3 ln | u | + C = 3 ln | x 10 | + C , x 10 .

See [link] .

A graph of the function f(x) = 3 / (x – 10). There is an asymptote at x=10. The first segment is a decreasing concave down curve that approaches 0 as x goes to negative infinity and approaches negative infinity as x goes to 10. The second segment is a decreasing concave up curve that approaches infinity as x goes to 10 and approaches 0 as x approaches infinity.
The domain of this function is x 10 .
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Find the antiderivative of 1 x + 2 .

ln | x + 2 | + C

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Finding an antiderivative of a rational function

Find the antiderivative of 2 x 3 + 3 x x 4 + 3 x 2 .

This can be rewritten as ( 2 x 3 + 3 x ) ( x 4 + 3 x 2 ) −1 d x . Use substitution. Let u = x 4 + 3 x 2 , then d u = 4 x 3 + 6 x . Alter du by factoring out the 2. Thus,

d u = ( 4 x 3 + 6 x ) d x = 2 ( 2 x 3 + 3 x ) d x 1 2 d u = ( 2 x 3 + 3 x ) d x .

Rewrite the integrand in u :

( 2 x 3 + 3 x ) ( x 4 + 3 x 2 ) −1 d x = 1 2 u −1 d u .

Then we have

1 2 u −1 d u = 1 2 ln | u | + C = 1 2 ln | x 4 + 3 x 2 | + C .
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Finding an antiderivative of a logarithmic function

Find the antiderivative of the log function log 2 x .

Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have

log 2 x d x = x ln 2 ( ln x 1 ) + C .
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Find the antiderivative of log 3 x .

x ln 3 ( ln x 1 ) + C

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[link] is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration.

Evaluating a definite integral

Find the definite integral of 0 π / 2 sin x 1 + cos x d x .

We need substitution to evaluate this problem. Let u = 1 + cos x , , so d u = sin x d x . Rewrite the integral in terms of u , changing the limits of integration as well. Thus,

u = 1 + cos ( 0 ) = 2 u = 1 + cos ( π 2 ) = 1 .

Then

0 π / 2 sin x 1 + cos x = 2 1 u −1 d u = 1 2 u −1 d u = ln | u | | 1 2 = [ ln 2 ln 1 ] = ln 2.
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Key concepts

  • Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
  • Substitution is often used to evaluate integrals involving exponential functions or logarithms.

Questions & Answers

find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
mujahid
How to do basic integrals
dondi Reply
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Jug
discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
ok ok hehe thanks nice dp ekko hahaha
Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
8x+12
Dhruv
8x+3
okhiria
d the derivative of y= e raised to power x
okhiria

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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