Integration by Change of Variable
A Dominican trainer uses a function to model how fast the volume of air in a baseball player's lungs changes during a workout. To find out how much air moved in total, the trainer needs an antiderivative, but the function is a composite expression, with a square root and a logarithm together. How do you integrate something like that? With a trick: give a name to whatever is getting in the way.
The chain rule backwards: recognizing g(u(x))·u'(x)
The chain rule says that $\big(G(u(x))\big)' = G'(u(x))\cdot u'(x)$. Read backwards: 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 $t = x^2 + 3$:
$$\int 2x \times (x^2 + 3)^4 \, dx$$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:
$$\int \frac{e^x}{e^x + 4} \, dx$$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.
Linear change: u = ax + b
The simplest change occurs when $ax+b$ appears inside a function. If $u=ax+b$, then $du=a\,dx$, that is, $dx=\dfrac{du}{a}$.
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 $t = 2x + 1$:
$$\int \sqrt{2x + 1} \, dx$$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.
The explicit change, step by step
When the structure isn't visible at a glance, you write $u=u(x)$ explicitly, solve for $dx$, and substitute everything, including the $dx$, in terms of $u$.
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 $t = e^x$:
$$\int \frac{e^{2x}}{e^x + 1} \, dx$$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 $t = \cos x$:
$$\int \frac{\sin x}{1 + \cos^2 x} \, dx$$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 $t = \ln x$:
$$\int \frac{1}{x(1 + \ln^2 x)} \, dx$$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:
$$\int \cos^3 x \, dx$$Changes with roots and logarithms
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 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 $u = \ln t$:
$$\int \frac{1}{t \sqrt{\ln t}} \, dt$$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.
More elaborate changes
Sometimes the change produces an expression that has to be transformed, or you have to chain two changes. The idea is the same: eliminate what's in the way, compute $dx$ carefully, and integrate.
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 $f(x) = \frac{\sqrt{x}}{1+\sqrt[3]{x}}$.
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$$Make the appropriate substitution to transform this trigonometric integral into a rational integral and solve it by partial fraction decomposition.
Find the antiderivative of the function $f(x) = \frac{e^{2x}}{\sqrt{e^x + 1}}$ that passes through the point $(0, 0)$. To do this, use the substitution $t^2 = e^x + 1$…
Find the antiderivative of the function $f(x) = \frac{e^{2x}}{\sqrt{e^x + 1}}$ that passes through the point $(0, 0)$. To do this, use the substitution $t^2 = e^x + 1$ and remember to reverse the substitution at the end of the process.
The velocity of a reconnaissance drone is given by the expression $v(t) = \frac{1}{t(1+\ln^2 t)}$, where $t$ is time in seconds ($t > 0$). If it is known that at the…
The velocity of a reconnaissance drone is given by the expression $v(t) = \frac{1}{t(1+\ln^2 t)}$, where $t$ is time in seconds ($t > 0$). If it is known that at the initial measurement time ($t=1$) the drone was at position $s=0$, obtain the position function $s(t)$ using the substitution $u = \ln t$.
Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$. Suggestion: Make the substitution $\sqrt{e^x - 1} = t$. You will need to solve for $x$ and calculate its…
Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$.
Suggestion: Make the substitution $\sqrt{e^x - 1} = t$. You will need to solve for $x$ and calculate its differential $dx$ carefully.
A team of architects is modelling the curvature of a new tunnel whose cross-section corresponds to the integral: $$\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$$Calculate the general antiderivative using the substitution $t = \sqrt{x+1}$.
Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges: $$\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.
Back to the start: for the air volume with $\frac{1}{x\sqrt{1+\ln x}}$, the change $u=1+\ln x$ eliminates the root and leaves $2\sqrt{1+\ln x}+C$.