Exercises: Integration by substitution

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 $t = 3x - 5$:

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 $t = 2x + 1$:

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 $u = \frac{x}{2}$:

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 $t = 4x$:

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.

Sometimes, substitution helps us simplify expressions where a function and its derivative appear multiplied. Solve the following integral by choosing the substitution $t = x^2 + 3$:

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.

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:

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 $t = \cos x$:

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 $t = \ln x$:

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:

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:

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 $u = \ln t$:

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:

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.

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.

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 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 differential $dx$ carefully.

A team of architects is modelling the curvature of a new tunnel whose cross-section corresponds to the integral:

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:

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