Question

Evaluate the following integrals: a. \(\int \sinh ^{3} x \cosh x d x\) b. \(\int \operatorname{sech}^{2}(3 x) d x\)

Short answer

a. \( \frac{\sinh^4 x}{4} + C \); b. \( \frac{1}{3} \tanh(3x) + C \)."

Step-by-step solution

Identify substitution for integral a

For the integral \( \int \sinh^3 x \cosh x \, dx \), we notice that \( \cosh x \, dx \) is the derivative of \( \sinh x \). Thus, use substitution: let \( u = \sinh x \), then \( du = \cosh x \, dx \). This transforms the integral to \( \int u^3 \, du \).

Evaluate the transformed integral a

The integral \( \int u^3 \, du \) can be solved using the power rule for integration. Compute it as follows: \[ \int u^3 \, du = \frac{u^4}{4} + C \].

Substitute back for integral a

Substitute back \( u = \sinh x \) into the result: \( \frac{(\sinh x)^4}{4} + C \). This simplifies to \( \frac{\sinh^4 x}{4} + C \).

Identify structure for integral b

For \( \int \operatorname{sech}^2(3x) \, dx \), notice that the derivative of \( \tanh(3x) \) is \( 3 \operatorname{sech}^2(3x) \). Use this knowledge to find the integral.

Apply substitution for integral b

Let \( u = 3x \), then \( du = 3 \, dx \) or \( dx = \frac{1}{3} \, du \). Transform the integral to \( \frac{1}{3} \int \operatorname{sech}^2(u) \, du \).

Evaluate the integral of secant squared

The integral \( \int \operatorname{sech}^2(u) \, du = \tanh(u) + C \). Thus, \( \frac{1}{3} \int \operatorname{sech}^2(u) \, du = \frac{1}{3} \tanh(u) + C \).

Substitute back for integral b

Substitute \( u = 3x \) back into the result: \( \frac{1}{3} \tanh(3x) + C \).

Key concepts

Hyperbolic Functions

Hyperbolic functions, like those in trigonometry, have their own set of identities and properties. Two of the most commonly used hyperbolic functions are \( \sinh x \) and \( \cosh x \). These are analogous to the sine and cosine functions but are derived from the exponential function \( e^x \). Specifically, \( \sinh x = \frac{e^x - e^{-x}}{2} \) and \( \cosh x = \frac{e^x + e^{-x}}{2} \). The derivative of \( \sinh x \) is \( \cosh x \), and the derivative of \( \cosh x \) is \( \sinh x \). This is important because these derivatives play key roles in solving integrals involving hyperbolic functions. Knowing these derivatives helps us identify good substitution candidates when integrating. Unlike trigonometric functions, hyperbolic functions are not periodic but exhibit exponential growth. This unique behavior makes them particularly useful in various applications like physics, particularly in hyperbolic geometry and the theory of relativity.

Integration Techniques

Integration techniques are methods used to solve integrals, especially when they are not immediately straightforward. One fundamental technique is the power rule, which states: if \( \,\int x^n \, dx \), then the integral is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Another important technique is using known anti-derivatives, which helps simplify the process. For example, the anti-derivative of \( \operatorname{sech}^2(x) \) is \( \tanh(x) \), because the derivative of \( \tanh(x) \) yields \( \operatorname{sech}^2(x) \). This makes evaluating certain integrals more straightforward. When faced with complicated integrals, breaking them down into simpler components or recognizing known forms can make the problem easier. When dealing with hyperbolic functions, using their properties and derivatives often offers good opportunities for simplification and solution.

Substitution Method

The substitution method, also known as \( u \)-substitution, is a powerful technique in integration. It's akin to the reverse of the chain rule in differentiation. The main idea is to substitute part of the integrand with a single variable \( u \) to simplify the integral into a form that is easier to solve. In our first example, \( \int \sinh^3 x \cosh x \, dx \), \( \sinh x \) is substituted with \( u \), making \( \cosh x \, dx \) the differential \( du \). This transforms the integral into \( \int u^3 \, du \), which is manageable using the power rule. For our second example, \( \int \operatorname{sech}^2(3x) \, dx \), we use \( u = 3x \). This reduces the complexity by converting \( dx \) into \( \frac{1}{3} \, du \) and the integral into a form involving \( \operatorname{sech}^2(u) \), where we know the anti-derivative. Substitution can greatly simplify integrals that involve compositions of functions or repeated patterns.