Question

Evaluate each integral. $$\int \tanh ^{2} x d x(\text {Hint: Use an identity. })$$

Short answer

Question: Evaluate the integral \(\int \tanh^2 x\, dx\). Answer: The integral \(\int \tanh^2 x\, dx = \tanh x - x + C\), where \(C\) is the constant of integration.

Step-by-step solution

Use the identity

Recall the identity \(\tanh^2 x = \operatorname{sech}^2 x - 1\). Rewrite the integral using this identity: $$\int \tanh^2 x\, dx = \int (\operatorname{sech}^2 x - 1)\, dx$$.

Integrate each term separately

Integrate the two terms of the integral separately: $$\int (\operatorname{sech}^2 x - 1)\, dx = \int \operatorname{sech}^2 x\, dx - \int 1\, dx$$

Integrate the first term

The antiderivative of \(\operatorname{sech}^2 x\) is actually the derivative of the hyperbolic tangent function, so the integral is simply: $$\int \operatorname{sech}^2 x\, dx = \tanh x$$

Integrate the second term

The antiderivative of \(1\) is simply \(x\), so the integral is: $$\int 1\, dx = x$$

Combine the results

Now, combine the results of steps 3 and 4: $$\int \tanh^2 x\, dx = \tanh x - x$$

Add the constant of integration

In the final step, add the constant of integration, \(C\): $$\int \tanh^2 x\, dx = \tanh x - x + C$$

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. Just as trigonometric functions arise from considering the unit circle, hyperbolic functions arise from considering the standard hyperbola \(x^2 - y^2 = 1\). The two most fundamental hyperbolic functions are the hyperbolic sine \(\sinh x\) and hyperbolic cosine \(\cosh x\), defined by the exponential functions: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\).

From these definitions, other hyperbolic functions like the hyperbolic tangent \(\tanh x\), which we encounter in our integral problem, are derived: \(\tanh x = \frac{\sinh x}{\cosh x}\). These functions often appear in various areas of mathematics and physics, including calculus, complex analysis, and relativistic physics.

Indefinite Integrals

In calculus, an indefinite integral, also known as an antiderivative, represents a function whose derivative is the given function. It is symbolized by the integral sign \(\int\) followed by a function and an infinitesimal increment \(dx\) which represents the variable of integration. However, since derivatives are not uniquely invertible, the indefinite integral is not a single function, but a family of functions that differ by a constant, as any function plus a constant has the same derivative. Therefore, when we find an indefinite integral, we include a constant of integration \(C\) at the end of our expression to account for all possible antiderivatives.

Integration Techniques

There are various techniques to compute integrals, often depending on the form of the function we're integrating. Some common techniques include substitution, integration by parts, and partial fraction decomposition. For our problem, the hyperbolic identity \(\tanh^2 x = \operatorname{sech}^2 x - 1\) simplifies the function, allowing us to use basic integration rules rather than more complex methods. When applying integration techniques, it's crucial to recognize patterns and identities that could simplify the integration process, as done with the hyperbolic identity in this instance.

Antiderivatives

An antiderivative of a function \(f(x)\) is a different function \(F(x)\) such that \(F'(x) = f(x)\). Finding an antiderivative is the inverse process of taking a derivative. In the case of our exercise, we are looking for the antiderivative of \(\tanh^2 x\). By using the identity for \(\tanh^2 x\), we convert it into a form where we can easily determine the antiderivatives of the resulting terms \(\operatorname{sech}^2 x\) and \(1\), which are \(\tanh x\) and \(x\) respectively. As in all indefinite integrals, we include a constant of integration \(C\) to account for the infinitely many antiderivatives that could fit the original function.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold for all values of the variables where both sides of the equality are defined. These identities have their hyperbolic analogs, which are useful in hyperbolic function integrations. In solving our integral of \(\tanh^2 x\), we rely on such an identity, transforming the integrand into a difference of \(\operatorname{sech}^2 x\) and \(1\), a step that makes the integration straightforward. The use of identities is a common and powerful technique in integration, as it can transform complex problems into simpler ones or convert them into forms where known antiderivatives exist.