Question

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

Short answer

Question: Evaluate the integral \(\int \sinh^2(z) dz\). Answer: \(\int \sinh^2(z) dz = \frac{1}{2}\sinh(2z) - \frac{1}{2}z + C\)

Step-by-step solution

Apply the identity to the integrand

First, we need to rewrite the given integral using the identity provided. We have: $$\int \sinh^2(z) dz = \int \frac{1}{2}(\cosh(2z) - 1) dz$$

Separate the integral into two

Now we need to separate the integral into two parts: $$\int \frac{1}{2}(\cosh(2z) - 1) dz = \frac{1}{2} \int (\cosh(2z) - 1) dz = \frac{1}{2}\int \cosh(2z) dz - \frac{1}{2}\int 1 dz$$

Integrate each part

We will now integrate each part separately: 1. Integrate the first part: \(\int \cosh(2z) dz\) The antiderivative of \(\cosh(x)\) is \(\sinh(x)\). So we will have $$\int \cosh(2z) dz = \frac{1}{2}\sinh(2z) + C_1$$ 2. Integrate the second part: \(\int 1 dz\) The antiderivative of \(1\) is simply the variable with respect to which we are integrating. In this case, we have: $$\int 1 dz = z + C_2$$

Combine the results and write the final answer

Now, we need to combine the results of the two parts and write the final answer. We have: $$\int \sinh^2(z) dz = \frac{1}{2}\sinh(2z) - \frac{1}{2}z + C$$ Here, \(C = C_1 + C_2\) is the constant of integration.

Key concepts

Hyperbolic Functions

Hyperbolic functions like sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), and others are analogs of the trigonometric functions but for a hyperbola, not a circle. They are defined using exponential functions. For instance, \(\sinh(x) = \frac{e^x - e^{-x}}{2}\) and \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). These functions have properties that closely resemble, but are distinct from, their circular counterparts.

For example, just as \(\sin^2(x) + \cos^2(x) = 1\) is a fundamental identity for sine and cosine, there’s an equivalent for hyperbolic functions: \(\cosh^2(x) - \sinh^2(x) = 1\). Importantly, knowledge of such identities is crucial in solving integration problems involving hyperbolic functions, as they often simplify the integrand into a more manageable form.

Integration by Substitution

Integration by substitution, also known as u-substitution, is a technique that simplifies the integration process by changing the variable of integration to a new variable. This method is comparable to the reverse process of the Chain Rule in differentiation and is particularly useful when dealing with composite functions. The basic idea is to substitute a part of the integrand with a new variable, differentiate this substitution to find a connection with the differential in the original integral, and then integrate with respect to the new variable.

After finding the antiderivative, we substitute back to the original variable. This method often transforms a complicated or unfamiliar integral into a simpler form that can be easily evaluated. It is a fundamental technique in calculus and one of the first strategies to consider when facing an integration problem.

Definite Integral Calculus

Definite integrals calculate the net area under a function's curve between two specified points along the x-axis. This means that rather than finding the general antiderivative, we are computing the accumulation of something, like area or total change, from point a to point b. The definite integral has upper and lower limits on the integral sign that indicate these bounds.

The process of finding a definite integral usually involves calculating an indefinite integral (also called an antiderivative) first and then applying the Fundamental Theorem of Calculus. This theorem relates the definite integral of a function to its antiderivative: by taking the difference of the antiderivative evaluated at the upper limit and the lower limit, we obtain the value of the definite integral.

Antiderivatives

Antiderivatives are the inverse process of differentiation. If a function F(x) is the antiderivative of f(x), then the derivative of F(x) equals f(x). The process of finding the antiderivative is known as integration. Since differentiation is a specific operation with a unique result, antiderivatives are not unique - a constant can always be added to them without changing the derivative. This constant is represented by C, called the constant of integration.

When integrating functions, such as hyperbolic functions or others involving a series of calculations, being familiar with basic antiderivatives and rules of integration is crucial. For instance, integrals of simple functions are often used as building blocks for more complex integrals. Understanding and recognizing patterns in integrals goes a long way in mastering the art of integration.