The Indefinite Integral. Antiderivatives and Basic Integrals
Wilkin is riding a guagua along the Autopista Duarte, and his cell phone records the speed at every instant. With that data he wants to know how far he has traveled. Until now you always went from a function $f$ to its derivative $f'$. Here the problem is the opposite: you know $f'$ and have to recover $f$. It is the same table of derivatives, but read in the opposite direction.
Differentiating backward: what an antiderivative is
If $s(t)$ is the position of the guagua and $v(t)=s'(t)$ is its velocity, recovering $s(t)$ from $v(t)$ is exactly this problem. Integrating is the opposite operation of differentiating.
In the graph, the parabola $x^2$ goes down while $2x$ is negative and goes up when $2x$ is positive. The line $2x$ is the derivative of the parabola.
Determine whether the function $F(x) = x \times e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental…
Determine whether the function $F(x) = x \times e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental relationship between a function and its antiderivative via differentiation.
Infinitely many antiderivatives: the constant $C$
To keep one specific antiderivative, one extra piece of data is enough: the value of $F$ at a point (an initial condition).
Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$. Note: First calculate the indefinite integral to obtain the…
Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$.
Note: First calculate the indefinite integral to obtain the family of functions with the constant $C$, and then use the point $P$ to find the specific value of that constant.
A professional skater is analyzing the trajectory of his latest jump on a ramp. The function that describes the variation of his height with respect to time (vertical…
A professional skater is analyzing the trajectory of his latest jump on a ramp. The function that describes the variation of his height with respect to time (vertical velocity) is given by the expression $v(t) = 3t^2 - 4t + 5$, whose graph appears in the figure. Find the family of functions $H(t)$ that represent the skater's height, by calculating the indefinite integral:
$$\int (3t^2 - 4t + 5) \, dt$$The table of derivatives, read backward
Each differentiation rule, read from right to left, gives a basic integral. There is no new table to memorize.
Why the one for $a^x$? Because $(a^x)' = a^x\ln a$, so you have to divide by $\ln a$ to get $a^x$.
In a study on how news spreads on a social network, it was determined that the growth rate of "shares" follows a power law. To find the cumulative total, a programmer…
In a study on how news spreads on a social network, it was determined that the growth rate of "shares" follows a power law. To find the cumulative total, a programmer must solve the following integral of an irrational function. Calculate it by first expressing the root as a fractional exponent:
$$\int \left( \sqrt[3]{x^2} + \frac{1}{\sqrt{x}} \right) \, dx$$In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the…
In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the expression for the function $g(x)$ by evaluating its indefinite integral:
$$\int \left( \frac{7}{1+x^2} - 2^x \right) \, dx$$Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:…
Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:
$f(x) = \dfrac{3}{x^4} - \dfrac{2}{\sqrt[3]{x^2}} + \dfrac{1}{5x}$
Linearity: sums, constants, and fractions
When the integrand is a fraction, there is no rule for the quotient. Prepare it:
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If the denominator is a monomial, split it into a sum of fractions.
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If the denominator is a polynomial and the degree of the numerator is greater than or equal to it, do the division.
Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and…
Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and splitting sums):
$$\int (4 \sin x - 5 \cos x + 3) \, dx$$Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.
Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.
Find the antiderivative of the function $f(x) = \dfrac{x^2 - 4x + 1}{x-2}$ whose graph passes through the point $P(3, 5)$. Note: Perform polynomial division first.
Find the antiderivative of the function $f(x) = \dfrac{x^2 - 4x + 1}{x-2}$ whose graph passes through the point $P(3, 5)$.
Note: Perform polynomial division first.
Functions of the form $f(ax+b)$
If the table says $\int \cos x\,dx=\sin x+C$, what happens with $\int\cos(5x)\,dx$? Try $\sin(5x)$: its derivative is $5\cos(5x)$, an extra 5. You have to divide by 5.
Examples of use:
| Integral | Result |
|---|---|
| $\int e^{3x-1}dx$ | $\frac{1}{3}e^{3x-1}+C$ |
| $\int \cos(5x)\,dx$ | $\frac{1}{5}\sin(5x)+C$ |
| $\int \frac{dx}{\cos^2(3x)}$ | $\frac{1}{3}\tan(3x)+C$ |
| $\int \frac{dx}{1+4x^2}$ | $\frac{1}{2}\arctan(2x)+C$ |
| $\int 5^{2x}dx$ | $\frac{5^{2x}}{2\ln 5}+C$ |
In the second-to-last one, $1+4x^2 = 1+(2x)^2$, so $a=2$. In the last one, the $\ln 5$ from the table combines with the $2$ from the argument.
A hearing study for teenagers analyses the intensity of a sound signal. The function describing the instantaneous sound pressure is $p(x) = \frac{3}{x} + e^{2x}$. Find…
A hearing study for teenagers analyses the intensity of a sound signal. The function describing the instantaneous sound pressure is $p(x) = \frac{3}{x} + e^{2x}$. Find the general antiderivative of this function by carrying out the integration:
$$\int \left( \frac{3}{x} + e^{2x} \right) \, dx$$Find the antiderivative $F(x)$ of the function $f(x) = \dfrac{1}{\cos^2(3x)} + 5^{2x}$ knowing that $F(0) = 1$.
Find the antiderivative $F(x)$ of the function $f(x) = \dfrac{1}{\cos^2(3x)} + 5^{2x}$ knowing that $F(0) = 1$.
Calculate the indefinite integral of the following function using linearity decomposition and adjusting for the derivative of the exponent:…
Calculate the indefinite integral of the following function using linearity decomposition and adjusting for the derivative of the exponent:
$I = \displaystyle\int \left( 2x \times e^{x^2+3} - \dfrac{5}{1+4x^2} \right) \, dx$
The chain rule in reverse: $f(u)\cdot u'$
It is the same idea with a simple inner function. Since $\big(F(u(x))\big)'=F'(u(x))\cdot u'(x)$, reading it backward:
Extension: roots, identities, and compositions
Determine the indefinite integral of the following composite function, identifying the derivative of the inner function:…
Determine the indefinite integral of the following composite function, identifying the derivative of the inner function:
$I = \displaystyle\int \dfrac{\ln^2(x) + 1}{x} \, dx$
Given the function $f(x) = \dfrac{\sin(x)}{\sqrt{1 + \cos(x)}}$, calculate its general antiderivative. Check your result by differentiating the obtained function.
Given the function $f(x) = \dfrac{\sin(x)}{\sqrt{1 + \cos(x)}}$, calculate its general antiderivative. Check your result by differentiating the obtained function.
Find the family of antiderivatives of the following trigonometric function using fundamental identities to transform the integrand into a sum of standard integrals:…
Find the family of antiderivatives of the following trigonometric function using fundamental identities to transform the integrand into a sum of standard integrals:
$$f(x) = \frac{\cos^3 x}{\sin^4 x}$$Logarithms: when the numerator is the derivative of the denominator
One case of the chain rule shows up all the time. If $f(u)=\frac{1}{u}$, then:
If the numerator is not an exact multiple of the derivative of the denominator, you split it into two fractions: one of the type $u'/u$ and another with a constant number in the numerator (which usually gives an arctangent).
Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:…
Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:
$I = \displaystyle\int \dfrac{3x - 1}{x^2 - \frac{2}{3}x + 5} \, dx$
Calculate the indefinite integral of the function: $$f(x) = \frac{\ln(\ln x)}{x \times \ln x}$$ This exercise assesses the ability to recognize the structure…
Calculate the indefinite integral of the function:
$$f(x) = \frac{\ln(\ln x)}{x \times \ln x}$$This exercise assesses the ability to recognize the structure $g(u(x)) \times u'(x)$ when there are multiple composite functions.
Preparing the integrand: completing the square and algebraic tricks
Some integrals are not in the table at first glance, but they will be after an algebraic change.
Arctangent. If the denominator is a square plus a constant, complete the square and reach $\frac{1}{1+w^2}$.
Arcsine. If there is a $\sqrt{a^2-(\cdot)^2}$, the formula is $\int\frac{dx}{\sqrt{a^2-x^2}}=\arcsin\frac{x}{a}+C$.
If the radicand is $4-(x-3)^2$, the arcsine formula is used with $x-3$ in place of $x$ and $a=2$. For a numerator like $x+1$ it is convenient to write it as $(x-3)+4$: the term $(x-3)$ is integrated with the chain rule (with $u=4-(x-3)^2$) and the $4$ gives an arcsine of $\frac{x-3}{2}$.
Exponentials in fractions. When the denominator has $e^x$, multiply numerator and denominator by $e^{-x}$ or divide first.
For fractions like $\frac{e^{2t}+3e^t}{e^t+1}$ you combine this with division: letting $w=e^t$, $\frac{w^2+3w}{w+1}=w+2-\frac{2}{w+1}$. Then $\frac{2}{w+1}=\frac{2}{e^t+1}$ is integrated as in the previous example.
Find the indefinite integral of the following rational function by completing the square in the denominator to identify an arctangent form: $$f(x) = \frac{3}{2x^2 - 4x +…
Find the indefinite integral of the following rational function by completing the square in the denominator to identify an arctangent form:
$$f(x) = \frac{3}{2x^2 - 4x + 10}$$A logistics company studies the tire wear of its fleet. The wear rate $V(x)$ (in mm per 1000 km) depends on the load $x$ (in metric tons) according to the function:…
A logistics company studies the tire wear of its fleet. The wear rate $V(x)$ (in mm per 1000 km) depends on the load $x$ (in metric tons) according to the function:
$V(x) = \frac{x + 1}{\sqrt{4 - (x - 3)^2}}$
Find the accumulated wear function $D(x)$, knowing that for a load of $3$ metric tons ($x = 3$) the wear is zero.
A sound engineer is analysing the signal of a new synthesiser. The rate of change of the wave intensity with respect to time, $t$, is given by the function…
A sound engineer is analysing the signal of a new synthesiser. The rate of change of the wave intensity with respect to time, $t$, is given by the function $f'(t) = \dfrac{e^{2t} + 3e^t}{e^t + 1}$.
Knowing that for a time $t = 0$ the intensity of the wave is $10$ units, determine the algebraic expression for the intensity $f(t)$ by performing the necessary algebraic manipulations to solve the integral.
Initial conditions and checking by differentiation
If you are given $f''$, you integrate twice and a constant appears each time. You need two pieces of data to pin them down.
When the integrand is hard to integrate but you have a candidate form, you can set up the antiderivative with coefficients and differentiate.
When the integrand is the derivative of a quotient
Some rational fractions with powers in the denominator, like $\dfrac{12x^2-4}{(x^2+1)^3}$, are not of the form $f(u)\,u'$. But they come from differentiating a quotient. If the denominator is $(x^2+1)^3$, the antiderivative must have denominator $(x^2+1)^2$ (when differentiating, the power goes up by one). So try $P\,x/(x^2+1)^2$ and adjust $P$.
With this method, the antiderivative of $\dfrac{12x^2-4}{(x^2+1)^3}=-4\cdot\dfrac{1-3x^2}{(x^2+1)^3}$ is $\dfrac{-4x}{(x^2+1)^2}$ plus a constant. Afterward, the second step is $\int\dfrac{-4x}{(x^2+1)^2}dx$, which is already of the chain type with $u=x^2+1$.
A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the…
A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the drone is at position $s=0$, determine the expression for the position function $s(t)$.
Determine the function $f(x)$ whose second derivative is $f''(x) = \dfrac{12x^2 - 4}{(x^2 + 1)^3}$. It is known that the graph of $f(x)$ has an inflection point with…
Determine the function $f(x)$ whose second derivative is $f''(x) = \dfrac{12x^2 - 4}{(x^2 + 1)^3}$.
It is known that the graph of $f(x)$ has an inflection point with positive $x$-coordinate at which the tangent line is horizontal, and that the graph of $f(x)$ passes through the point $(0, 5)$.
Note: This exercise requires an expert handling of the chain rule in reverse for rational functions.
Determine the values of the parameters $A, B$, and $C$ such that the function $F(x) = (Ax^2 + B) \sin(2x) + Cx \cos(2x)$ is an antiderivative of $f(x) = x^2 \cos(2x)$.…
Determine the values of the parameters $A, B$, and $C$ such that the function $F(x) = (Ax^2 + B) \sin(2x) + Cx \cos(2x)$ is an antiderivative of $f(x) = x^2 \cos(2x)$.
Set up the system of equations by differentiating $F(x)$ and equating coefficients.
One more step: product of functions
On each piece you integrate and the constants are adjusted so that the antiderivative is continuous. For example, $|x-2|e^x$ is $(x-2)e^x$ if $x\ge2$ and $-(x-2)e^x$ if $x<2$. With $\int(x-2)e^x\,dx=(x-3)e^x+C$ (you can check it by differentiating), the constant of each piece is chosen so that both agree at $x=2$.
Given the function $f(x) = |x - 2| \times e^x$, find the antiderivative $F(x)$ satisfying $F(0) = 1$. Remember that for $F(x)$ to be an antiderivative on its entire…
Given the function $f(x) = |x - 2| \times e^x$, find the antiderivative $F(x)$ satisfying $F(0) = 1$.
Remember that for $F(x)$ to be an antiderivative on its entire domain, it must be a continuous function, so you will need to adjust the constants of integration across the different intervals.
Go back to the beginning: with the velocity $v(t)$ recorded by Wilkin's cell phone, the distance traveled is an antiderivative $s(t)$ with $s'(t)=v(t)$. The constant $C$ is the starting point: it is fixed with an initial piece of data.