Integration by change of variable
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 variable is simply giving a name to that $u(x)$ so that you don't lose sight of it.
The chain rule in reverse: recognising $g(u(x))\cdot u'(x)$
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 $t = 3x - 5$:
$$\int (3x - 5)^7 \, dx$$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 $u=u(x)$ down 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 $t = \cos x$:
$$\int \frac{\sin x}{1 + \cos^2 x} \, dx$$Changes with roots and composite functions
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.
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:
$$\int \frac{\sqrt{x}}{\sqrt[3]{x} + 1} \, dx$$