Question

For the following exercises, find the antiderivatives for the given functions.\(3 x^{3} \tanh \left(x^{4}\right)\)

Short answer

The antiderivative is \( \frac{3}{4} \ln(\cosh(x^4)) + C \).

Step-by-step solution

Rewrite the Integral

The objective is to find the antiderivative of the function \( 3 x^3 \tanh(x^4) \). To do this, we'll rewrite it as an integral that we'll solve: \[\int 3 x^3 \tanh(x^4) \, dx.\]

Identify and Apply a Substitution

Notice the composition of functions, suggesting that substitution might simplify the integral. Let \( u = x^4 \), which gives us \( du = 4x^3 \, dx \). Thus, \( x^3 \, dx = \frac{1}{4} \, du \). Substitute these into the integral:\[\int 3 \tanh(u) \cdot \frac{1}{4} \, du = \frac{3}{4} \int \tanh(u) \, du.\]

Integrate with Respect to the New Variable

Now, we need to find the antiderivative of \( \tanh(u) \). Recall that the integral of \( \tanh(u) \) is \( \ln(\cosh(u)) + C \). Therefore:\[\frac{3}{4} \int \tanh(u) \, du = \frac{3}{4} \big( \ln(\cosh(u)) + C \big).\]

Substitute Back to Original Variable

Replace \( u \, \) back with \( x^4 \) to express the antiderivative in terms of \( x \): \[\frac{3}{4} \ln(\cosh(x^4)) + C.\] This is the original antiderivative of the function \( 3 x^3 \tanh(x^4) \).

Key concepts

Tanh Function

The tanh function, short for hyperbolic tangent, is a mathematical function that resembles the tangent function from trigonometry, but it is defined using hyperbolic sine and cosine functions. Specifically, it is expressed as \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). Like the ordinary tangent function, tanh describes the relationship between certain calculations involving hyperbolic angles.
This function is particularly useful in calculus and physics, especially in scenarios involving hyperbolic geometry. Its graph resembles an "S" shape, saturating to -1 and 1, which makes it useful in various scientific applications like neural networks. Its antiderivative, useful in integral calculus, is expressed using the hyperbolic cosine function: \( \int \tanh(x) \ dx = \ln|\cosh(x)| + C \), where \( C \) is the constant of integration.

Substitution Method

The substitution method, also known as \( u \)-substitution, is a technique used in integral calculus to simplify the integration process. When an integral contains a composite function, substitution helps to transform it into a more manageable form.
Here's how the method works:

• Identify the inner function \( u \), which can simplify the integral when substituted. This is often a function inside another function or causing complexity.
• Differentiate \( u \) to find \( du \) in terms of \( dx \).
• Rewrite the entire integral in terms of \( u \) and \( du \). This often reduces the original integral to a simpler form.
• Perform the integration with respect to the new variable \( u \).
• Finally, substitute back in terms of the original variable to obtain the final antiderivative.

Using substitution, complex integrals become much more approachable. It simplifies the integration by transforming it to an easier problem.

Cosh Function

The cosh function, or hyperbolic cosine, is a fundamental concept in differential calculus and mathematics. It is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and is akin to the cosine function but within the realm of hyperbolic geometry.
Its properties make it symmetric, producing a smooth, "U"-shaped graph that is always positive. One key characteristic is that its rate of growth is exponential, similar to the exponential function, due to its definition.
In integral calculus, the cosh function appears when integrating the tanh function. The antiderivative of the tanh function involves the natural logarithm of the cosh function, as seen in \( \int \tanh(u) \, du = \ln(\cosh(u)) + C \). Here, cosh serves as the inversion element, allowing for easier integration of similar hyperbolic functions.

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integrals. It deals with finding the antiderivatives or the "integrals" of functions. The integral can be thought of as the inverse operation to differentiation.
There are two main types of integrals:

• Definite integrals, which compute the area under a curve between two specific points.
• Indefinite integrals, used to find a general form of antiderivative for a given function.

When solving problems involving finding antiderivatives, like in our original problem, we typically use indefinite integrals. Integral calculus is essential for various applications in physics, engineering, and other scientific fields, allowing for the calculation of areas, volumes, and other quantities that add up to a whole. Applying techniques like substitution helps in solving more complex integrals effectively.