Integration by Substitution
Priya differentiates $\sin(x^2)$ with the chain rule and gets $2x\cos(x^2)$. Then she asks you to go the other way: what is $\displaystyle\int x\cos(x^2)\,dx$? None of the basic antiderivative formulas fits, because the inside of the cosine is $x^2$, not $x$. The trick is to read the chain rule backward. We'll answer her question in the next section.
Reading the chain rule backward
The chain rule says that if $F' = f$, then
$$\frac{d}{dx}\Big[F\big(g(x)\big)\Big] = f\big(g(x)\big)\cdot g'(x).$$Run it in reverse and you have a way to find antiderivatives. (Some problems call an antiderivative a primitive; it is the same thing.)
The key to every pattern is to find an inside function $g(x)$ whose derivative $g'(x)$ also appears as a factor.
The graph shows $x\cos(x^2)$ together with $\tfrac12\sin(x^2)$. The slope of the second curve at every point equals the height of the first.
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 $t = x^2 + 3$:
$$\int 2x \cdot (x^2 + 3)^4 \, dx$$Determine the primitive of the function $f(x) = \frac{\ln(x)}{x}$ using the substitution $t = \ln(x)$. This exercise is fundamental for identifying patterns where the…
Determine the primitive 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.
The u-substitution method
When the pattern is not obvious at a glance, you write the substitution down. Naming the inside function $u$ lets you keep track of everything, including $dx$.
Now we can answer Priya's question from the start.
Calculate the primitive of the following function using the substitution $t = e^x$. Notice how the presence of the derivative of the exponential in the numerator…
Calculate the primitive of the following function using the substitution $t = e^x$. Notice how the presence of the derivative of the exponential in the numerator facilitates the resolution:
$$\int \frac{e^x}{e^x + 4} \, dx$$A team of sound engineers is analyzing the pressure wave of a new synthesizer. The function describing the energy accumulation is given by the integral of a composite…
A team of sound engineers is analyzing the pressure wave of a new synthesizer. The function describing the energy accumulation is given by the integral of a composite power. Calculate the following primitive 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. Find the primitive of the following function by making…
To calculate the area of a section of a curved solar panel, an architect needs to solve an irrational integral. Find the primitive of the following function by making the suggested substitution $t = 2x + 1$:
$$\int \sqrt{2x + 1} \, dx$$When the inside is linear
A very common case is an inside function of the form $u=ax+b$. Then $du=a\,dx$, so $dx=\dfrac{1}{a}\,du$. The constant $\tfrac1a$ just rides along.
In the design of a ramp for a skatepark, an exponential curve is used to smooth the transition. Find the family of primitive functions of the following expression using…
In the design of a ramp for a skatepark, an exponential curve is used to smooth the transition. Find the family of primitive functions of the following expression using the substitution $u = \frac{x}{2}$:
$$\int e^{x/2} \, dx$$A video game programmer is calculating the oscillatory trajectory of a projectile in an atmosphere with resistance. Determine the primitive of the following…
A video game programmer is calculating the oscillatory trajectory of a projectile in an atmosphere with resistance. Determine the primitive of the following trigonometric function using the substitution $t = 4x$:
$$\int \cos(4x) \, dx$$Calculate the primitive 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…
Calculate the primitive 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 integral in a more structured way.
Choosing u: logs, trig, and roots
Which piece should be $u$? Look for the inside function whose derivative is also in the integrand.
| If you see... | try $u=$ | because $du$ is... |
|---|---|---|
| $\dfrac{1}{x}$ together with $\ln x$ | $\ln x$ | $\dfrac{1}{x}\,dx$ |
| $\cos x$ together with a function of $\sin x$ | $\sin x$ | $\cos x\,dx$ |
| $\sin x$ together with a function of $\cos x$ | $\cos x$ | $-\sin x\,dx$ |
| a root $\sqrt{\ \cdot\ }$ of an expression | the expression under the root | its derivative times $dx$ |
| $e^x$ in the numerator | $e^x$ or the exponent | $e^x\,dx$ |
Two standard integrals show up constantly after a substitution:
$$\int u^{-1/2}\,du = 2\sqrt u + C \qquad\qquad \int\frac{du}{1+u^2}=\arctan u+C$$To analyze the intensity of an acoustic signal in decibels, a sound technician must find the integral function of the sound pressure. Calculate the following indefinite…
To analyze the intensity of an acoustic signal in decibels, a sound technician must find the integral function of the 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 any instant $t > 1$ is given by the function $v(t) = \frac{1}{t \sqrt{\ln t}}$. If it is known that position is the…
A surveillance drone moves such that its velocity at any instant $t > 1$ is given by the function $v(t) = \frac{1}{t \sqrt{\ln t}}$. If it is known that position is the integral of velocity, find the expression for the distance traveled using the substitution $u = \ln t$:
$$\int \frac{1}{t \sqrt{\ln t}} \, dt$$In the study of alternating current circuits, the need to integrate combined trigonometric functions frequently arises. Find the primitive of the following function by…
In the study of alternating current circuits, the need to integrate combined trigonometric functions frequently arises. Find the primitive of the following function by making the substitution $t = \cos x$:
$$\int \frac{\sin x}{1 + \cos^2 x} \, dx$$Leftover x: solve for x in terms of u
Sometimes $du$ does not account for every $x$ in the integrand, for example $\displaystyle\int\frac{x}{\sqrt{x+3}}\,dx$. The fix is to solve your substitution for $x$ and rewrite everything in the new variable.
In the design of a ramp for a skatepark, engineers have determined that the side profile follows the gradient of a function whose primitive requires solving the…
In the design of a ramp for a skatepark, engineers have determined that the side profile follows the gradient of a function whose primitive requires solving the following integral. Use the substitution $t^2 = x + 3$ to calculate the family of primitive functions of:
$$\int \frac{x}{\sqrt{x+3}} \, dx$$Find the family of primitives of the following rational function containing a square root in the denominator. It is suggested to use the substitution $t = \sqrt{x - 1}$…
Find the family of primitives 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 modeling 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 modeling the curvature of a new tunnel whose cross-section corresponds to the integral:
$$\int \frac{1}{1 + \sqrt{x+1}} dx$$Calculate the general primitive using the substitution $t = \sqrt{x+1}$.
Exponentials: $u=e^x$ and roots of $e^x\pm k$
When the integrand is a rational expression in $e^x$, let $t=e^x$. Then $dt=e^x\,dx=t\,dx$, so $dx=\dfrac{dt}{t}$, and $e^{2x}=t^2$.
If there is a root such as $\sqrt{e^x+k}$, let $t$ be the root itself. Then $t^2=e^x+k$, so $e^x=t^2-k$ and
$$x=\ln(t^2-k),\qquad dx=\frac{2t}{t^2-k}\,dt.$$A biological laboratory is studying the growth rate of a bacteria population. The growth function is given by an expression that requires integrating the following…
A biological laboratory is studying the growth rate of a bacteria population. The growth function is given by an expression that requires integrating the following quotient. Calculate the primitive using the substitution $t = e^x$:
$$\int \frac{e^{2x}}{e^x + 1} \, dx$$Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$. Hint: Use the substitution $\sqrt{e^x - 1} = t$. You will need to make $x$ the subject and calculate its…
Calculate the indefinite integral $\int \sqrt{e^x - 1} dx$.
Hint: Use the substitution $\sqrt{e^x - 1} = t$. You will need to make $x$ the subject and calculate its differential $dx$ carefully.
Trig integrals: odd powers and trig-to-rational
Two situations come up often.
Odd power of sine or cosine. Split off one factor and turn the rest into the other function with $\sin^2x+\cos^2x=1$. Then $u$ is the other function.
Trig function that becomes a rational function. If $\cos x\,dx$ is present and everything else is a function of $\sin x$, let $u=\sin x$. You get a rational integral that you can solve with partial fractions.
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 the design of a ramp for a skatepark, an engineer needs to find the primitive of the gradient function defined by: $$\int \frac{\cos x}{\operatorname{sen}^2 x -…
In the design of a ramp for a skatepark, an engineer needs to find the primitive of the gradient function defined by:
$$\int \frac{\cos x}{\operatorname{sen}^2 x - 5\operatorname{sen} x + 6} dx$$Make the appropriate substitution to transform this trigonometric integral into a rational integral and solve it using partial fractions.
Roots with different indices
If an integrand contains both $\sqrt{x}$ and $\sqrt[3]{x}$, one substitution has to remove both roots. Take $x=t^n$, where $n$ is the LCM of the indices. For indices 2 and 3, $n=6$, so $x=t^6$ and $dx=6t^5\,dt$. Then $\sqrt{x}=t^3$ and $\sqrt[3]{x}=t^2$.
Calculate the primitive of the following function with radicals of different indices. To do this, use a substitution of the type $x = t^n$, where $n$ is the lowest…
Calculate the primitive of the following function with radicals of different indices. To do this, use a substitution of the type $x = t^n$, where $n$ is the lowest common multiple of the indices of the roots:
$$\int \frac{\sqrt{x}}{\sqrt[3]{x} + 1} \, dx$$Determine the family of primitives of the function $f(x) = \frac{\sqrt{x}}{1+\sqrt[3]{x}}$. Hint: Use a substitution of the form $x = t^n$, where $n$ is the lowest…
Determine the family of primitives of the function $f(x) = \frac{\sqrt{x}}{1+\sqrt[3]{x}}$.
Hint: Use a substitution of the form $x = t^n$, where $n$ is the lowest common multiple of the indices of the roots present in the expression.
Antiderivatives through a given point
An indefinite integral gives a whole family of functions, one for each value of $C$. If the problem says the function passes through a point (or gives an initial position), plug that point in to find $C$.
Find the primitive 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…
Find the primitive 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 the 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 the time in seconds ($t > 0$). If it is known that at the initial instant of measurement ($t=1$) the drone was at position $s=0$, find the position function $s(t)$ using the substitution $u = \ln t$.
Nested and repeated substitutions
Some integrals need more than one substitution. After the first one, look again at the new integral. If it still has an inside function and its derivative, substitute again.
Calculate the indefinite integral of the following function, which represents the rate of change of the volume of air in an athlete's lungs during high-intensity…
Calculate the indefinite integral of the following function, which represents the rate of change of the volume of air in an athlete's lungs during high-intensity training:
$$\int \frac{1}{x \sqrt{1+\ln x}} dx$$To solve it, propose a substitution that eliminates the square root in the denominator.
Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges: $$\int \frac{\ln(\ln x)}{x \ln x} dx$$ Show step-by-step the two…
Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges:
$$\int \frac{\ln(\ln x)}{x \ln x} dx$$Show step-by-step the two successive substitutions you would make to simplify the expression until it becomes a standard integral.