Integration by change of variable

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:

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.

To calculate the family of antiderivatives of the rational function $f(x) = \frac{2x}{x^2 + 4}$, two students propose different approaches:

• Aoife argues that the integral is "almost immediate" because the numerator is exactly the derivative of the denominator, so she directly applies the formula $\int \frac{u'(x)}{u(x)}\, dx = \ln|u(x)| + C$.

• Seán is wary of solving it by inspection and prefers to formally apply the substitution method by setting $u = x^2 + 4$, finding $du$, and solving for $dx$ to transform the integral.

1. a) Work out the full solution following Aoife's procedure and also following Seán's.
2. b) Compare both methods and justify why Aoife's saves time in this specific case without risk of error, also indicating under what algebraic circumstances Seán's systematic procedure is safer.

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 $t = 3x - 5$:

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 $t = 4x$:

Cian has to calculate the antiderivative of the function $f(x) = (4x - 1)^5$ and writes on his exam:

justifying that he applied the power rule by adding 1 to the exponent and dividing by the new exponent.

1. a) Show, by differentiating the result Cian obtained, why his antiderivative is wrong.
2. b) Explain what essential element of the chain rule he forgot to compensate for in his work.
3. c) Find the correct antiderivative by adjusting the missing constant so that the derivative of the base appears, or alternatively, by applying the formal substitution $u = 4x - 1$.

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 $t = e^x$:

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 $t = \cos x$:

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:

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:

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:

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:

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 lowest common multiple of the root indices:

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:

Make the appropriate substitution to transform this trigonometric integral into a rational integral and solve it by partial fraction decomposition.

During a high-temperature episode recorded by Met Éireann in Cork, the additional power demanded from the electrical grid due to the intensive use of air conditioning systems evolves according to the instantaneous rate of change:

where $t$ is the time measured in hours elapsed since 12:00 ($0 \le t \le 6$) and $p(t)$ is expressed in megawatts per hour ($\text{MW/h}$).

1. a) Model the accumulated energy consumption function by calculating the family of antiderivatives $E(t) = \int p(t)\, dt$ using the substitution $u = t^2 + 16$.
2. b) Knowing that at 12:00 ($t = 0$) the accumulated energy meter read $240\text{ MWh}$, determine in a reasoned manner the value of the integration constant $C$ and write the particular function $E(t)$.
3. c) Using the function obtained, calculate how many additional megawatt-hours ($\text{MWh}$) the region has consumed between 12:00 ($t = 0$) and 15:00 ($t = 3$).

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 $u = \ln t$.

Solve the following integral combining nested logarithms, typical of advanced end-of-unit challenges:

Indicate step by step the two successive substitutions you would make to simplify the expression into a standard integral.