Integration by Change of Variable
A reconnaissance drone moves with velocity $v(t) = \dfrac{1}{t\sqrt{\ln t}}$ meters per second, with $t > 1$. To know where it is, you need an antiderivative of $v$, and that function is not in any table of basic integrals. The technique in this section turns it into an integral you do know how to solve.
Chain rule backward: recognizing $g(u(x))\cdot u'(x)$
The chain rule says that $\big(G(u(x))\big)' = G'(u(x))\cdot u'(x)$. Read backward: if the integrand has the form $g(u(x))\cdot u'(x)$, its antiderivative is $G(u(x))$, where $G$ is an antiderivative of $g$. Changing variables means giving a name to that $u(x)$ so you don't lose track of it.
Sometimes, substitution helps us simplify expressions where a function and its derivative appear multiplied. Solve the following integral by choosing the substitution…
Sometimes, substitution helps us simplify expressions where a function and its derivative appear multiplied. Solve the following integral by choosing the substitution $t = x^2 + 3$:
$$\int 2x \times (x^2 + 3)^4 \, dx$$Determine the antiderivative of the function $f(x) = \frac{\ln(x)}{x}$ using the substitution $t = \ln(x)$. This exercise is fundamental for identifying patterns where…
Determine the antiderivative of the function $f(x) = \frac{\ln(x)}{x}$ using the substitution $t = \ln(x)$. This exercise is fundamental for identifying patterns where the differential $dt$ consumes part of the original expression.
Calculate the antiderivative of the following function using the substitution $t = e^x$. Observe how the presence of the derivative of the exponential in the numerator…
Calculate the antiderivative of the following function using the substitution $t = e^x$. Observe how the presence of the derivative of the exponential in the numerator facilitates the solution:
$$\int \frac{e^x}{e^x + 4} \, dx$$Linear changes: $u = ax+b$
The simplest case is when what's inside the function is $ax+b$. Then $u' = a$ is a constant and you compensate by dividing by $a$.
A team of sound engineers is analysing the pressure wave of a new synthesiser. The function describing energy accumulation is given by the integral of a composite power.…
A team of sound engineers is analysing the pressure wave of a new synthesiser. The function describing energy accumulation is given by the integral of a composite power. Calculate the following antiderivative using the substitution $t = 3x - 5$:
$$\int (3x - 5)^7 \, dx$$To calculate the area of a section of a curved solar panel, an architect needs to solve an irrational integral. Obtain the antiderivative of the following function by…
To calculate the area of a section of a curved solar panel, an architect needs to solve an irrational integral. Obtain the antiderivative of the following function by making the suggested substitution $t = 2x + 1$:
$$\int \sqrt{2x + 1} \, dx$$In designing a ramp for a skatepark, an exponential curve is used to smooth the transition. Find the family of antiderivatives of the following expression using the…
In designing a ramp for a skatepark, an exponential curve is used to smooth the transition. Find the family of antiderivatives of the following expression using the substitution $u = \frac{x}{2}$:
$$\int e^{x/2} \, dx$$The explicit change, step by step
When the structure isn't visible at a glance, you write $u=u(x)$, solve for $dx$, and substitute everything in terms of $u$, including the $dx$.
A video game programmer is calculating the oscillatory trajectory of a projectile in an atmosphere with resistance. Determine the antiderivative of the following…
A video game programmer is calculating the oscillatory trajectory of a projectile in an atmosphere with resistance. Determine the antiderivative of the following trigonometric function using the substitution $t = 4x$:
$$\int \cos(4x) \, dx$$Find the antiderivative of the function $f(x) = \frac{1}{5x + 2}$ by making the substitution $t = 5x + 2$. Remember that this method will allow you to obtain a…
Find the antiderivative of the function $f(x) = \frac{1}{5x + 2}$ by making the substitution $t = 5x + 2$. Remember that this method will allow you to obtain a logarithmic-type integral in a more structured way.
A biological laboratory is studying the growth rate of a bacterial population. The growth function is given by an expression requiring the integration of the following…
A biological laboratory is studying the growth rate of a bacterial population. The growth function is given by an expression requiring the integration of the following quotient. Calculate the antiderivative using the substitution $t = e^x$:
$$\int \frac{e^{2x}}{e^x + 1} \, dx$$Logarithms and roots of a function
When $\ln x$ appears alongside a factor $\frac1x$, or a root whose inner expression has a derivative in plain sight, it's useful to call that inner expression $u$.
If something like $\ln(\ln x)$ appears inside the integral, you can make two changes in a row: first $u=\ln x$, and then, in the new integral, another change on whatever remains.
Back to the beginning: for the drone, $u=\ln t$ gives $du=\frac{dt}{t}$ and the integral becomes $\int u^{-1/2}\,du$, which is a basic power. Now you know how it's done.
To analyse the intensity of an acoustic signal in decibels, a sound engineer must find the integral function of sound pressure. Calculate the following indefinite…
To analyse the intensity of an acoustic signal in decibels, a sound engineer must find the integral function of sound pressure. Calculate the following indefinite integral by applying the substitution $t = \ln x$:
$$\int \frac{1}{x(1 + \ln^2 x)} \, dx$$A surveillance drone moves such that its velocity at each instant $t > 1$ is given by the function $v(t) = \frac{1}{t \sqrt{\ln t}}$. Knowing that position is the…
A surveillance drone moves such that its velocity at each instant $t > 1$ is given by the function $v(t) = \frac{1}{t \sqrt{\ln t}}$. Knowing that position is the integral of velocity, find the expression for the distance travelled using the substitution $u = \ln t$:
$$\int \frac{1}{t \sqrt{\ln t}} \, dt$$Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges: $$\int \frac{\ln(\ln x)}{x \ln x} dx$$ Indicate step by step the…
Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges:
$$\int \frac{\ln(\ln x)}{x \ln x} dx$$Indicate step by step the two successive substitutions you would make to simplify the expression into a standard integral.
Roots with $x$ outside: solving for $x$
Sometimes, besides the root, there's a stray $x$ in the numerator. Then the substitution is made for the root, and you also solve for $x$ to express it in terms of the new variable.
In designing a ramp for a skatepark, engineers have determined that the side profile follows the gradient of a function whose antiderivative requires solving the…
In designing a ramp for a skatepark, engineers have determined that the side profile follows the gradient of a function whose antiderivative requires solving the following integral. Use the substitution $t^2 = x + 3$ to find the family of antiderivatives of:
$$\int \frac{x}{\sqrt{x+3}} \, dx$$Find the family of antiderivatives of the following rational function containing a square root in the denominator. It is suggested to use the substitution…
Find the family of antiderivatives of the following rational function containing a square root in the denominator. It is suggested to use the substitution $t = \sqrt{x - 1}$ to simplify the expression:
$$\int \frac{x+2}{x \sqrt{x-1}} \, dx$$A team of architects is modelling the curvature of a new tunnel whose cross-section corresponds to the integral: $$\int \frac{1}{1 + \sqrt{x+1}} dx$$ Calculate the…
A team of architects is modelling the curvature of a new tunnel whose cross-section corresponds to the integral:
$$\int \frac{1}{1 + \sqrt{x+1}} dx$$Calculate the general antiderivative using the substitution $t = \sqrt{x+1}$.
Radicals of different index and roots of exponentials
If $\sqrt{x}$ and $\sqrt[3]{x}$ appear at the same time, use $x=t^n$ with $n$ equal to the LCM of the indices. That way all the roots become integer powers of $t$.
When the root contains an exponential, such as $\sqrt{e^x-1}$, let $t=\sqrt{e^x-1}$. Solving, $e^x=t^2+1$, so $x=\ln(t^2+1)$ and $dx=\dfrac{2t}{t^2+1}\,dt$. The integral turns into a rational one.
Calculate the antiderivative of the following function containing radicals of different indices. To do this, use a substitution of the type $x = t^n$, where $n$ is the…
Calculate the antiderivative of the following function containing radicals of different indices. To do this, use a substitution of the type $x = t^n$, where $n$ is the lowest common multiple of the root indices:
$$\int \frac{\sqrt{x}}{\sqrt[3]{x} + 1} \, dx$$Determine the family of antiderivatives of the function $f(x) = \frac{\sqrt{x}}{1+\sqrt[3]{x}}$.
Determine the family of antiderivatives of the function $f(x) = \frac{\sqrt{x}}{1+\sqrt[3]{x}}$.
Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$. Suggestion: Make the substitution $\sqrt{e^x - 1} = t$. You will need to solve for $x$ and calculate its…
Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$.
Suggestion: Make the substitution $\sqrt{e^x - 1} = t$. You will need to solve for $x$ and calculate its differential $dx$ carefully.
Changes in trigonometric integrals
If the integrand has $\sin x$ or $\cos x$, often the derivative of one is in the other. With $u=\cos x$ you have $du=-\sin x\,dx$; with $u=\sin x$, $du=\cos x\,dx$. For odd powers of sine or cosine, you split off one factor and write the rest using $\sin^2x+\cos^2x=1$.
Sometimes the change turns the trigonometric integral into a rational one, which is solved with partial fractions (see PRIM-RAC).
In the study of alternating current circuits, the need to integrate combined trigonometric functions frequently arises. Find the antiderivative of the following function…
In the study of alternating current circuits, the need to integrate combined trigonometric functions frequently arises. Find the antiderivative of the following function by making the substitution $t = \cos x$:
$$\int \frac{\sin x}{1 + \cos^2 x} \, dx$$Calculate the following indefinite integral using the substitution $t = \sin x$. Pay attention to how to transform the differential and the remaining powers of $\cos x$…
Calculate the following indefinite integral using the substitution $t = \sin x$. Pay attention to how to transform the differential and the remaining powers of $\cos x$ using the fundamental trigonometric identity:
$$\int \cos^3 x \, dx$$In designing a ramp for a skatepark, an engineer needs to find the antiderivative of the gradient function defined by: $$\int \frac{\cos x}{\operatorname{sen}^2 x -…
In designing a ramp for a skatepark, an engineer needs to find the antiderivative of the gradient function defined by:
$$\int \frac{\cos x}{\operatorname{sen}^2 x - 5\operatorname{sen} x + 6} dx$$Make the appropriate substitution to transform this trigonometric integral into a rational integral and solve it by partial fraction decomposition.
Fixing the constant $C$ with a condition
The general antiderivative has a free $C$. If the problem says the curve passes through a point, or that at a certain instant the drone is at a certain position, you substitute those data to fix $C$. Do it after undoing the change, with the expression in the original variable.
Find the antiderivative of the function $f(x) = \frac{e^{2x}}{\sqrt{e^x + 1}}$ that passes through the point $(0, 0)$. To do this, use the substitution $t^2 = e^x + 1$…
Find the antiderivative of the function $f(x) = \frac{e^{2x}}{\sqrt{e^x + 1}}$ that passes through the point $(0, 0)$. To do this, use the substitution $t^2 = e^x + 1$ and remember to reverse the substitution at the end of the process.
The velocity of a reconnaissance drone is given by the expression $v(t) = \frac{1}{t(1+\ln^2 t)}$, where $t$ is time in seconds ($t > 0$). If it is known that at the…
The velocity of a reconnaissance drone is given by the expression $v(t) = \frac{1}{t(1+\ln^2 t)}$, where $t$ is time in seconds ($t > 0$). If it is known that at the initial measurement time ($t=1$) the drone was at position $s=0$, obtain the position function $s(t)$ using the substitution $u = \ln t$.
Calculate the indefinite integral of the following function, which represents the rate of change of air volume in an athlete's lungs during high-intensity training:…
Calculate the indefinite integral of the following function, which represents the rate of change of air volume in an athlete's lungs during high-intensity training:
$$\int \frac{1}{x \sqrt{1+\ln x}} dx$$To solve it, propose a substitution that eliminates the square root in the denominator.