Question

Use integration by parts to show that $$\int \sec ^{3} x d x=\frac{1}{2} \sec x \tan x+\frac{1}{2} \int \sec x d x.$$

Short answer

Question: Prove that the integral of the secant cubed function is equal to $$\frac{1}{2}\sec x \tan x + \frac{1}{2} \int \sec x dx$$. Answer: To prove that the integral of the secant cubed function is equal to $$\frac{1}{2}\sec x \tan x + \frac{1}{2} \int \sec x dx$$, we applied the integration by parts method with $$u(x) = \sec x$$ and $$v'(x) = \sec^2 x$$. After simplification and solving for the original integral, we obtained the given expression, confirming that the statement is true.

Step-by-step solution

Identify u(x) and v'(x)

Choose u(x) to be $$\sec x$$ and v'(x) to be $$\sec^2 x$$. Choosing u(x) in this way will allow us to simplify the resulting integral when we find u'(x) and v(x).

Compute u'(x) and v(x)

Differentiate u(x) with respect to x to get u'(x): $$u'(x) = \frac{d}{dx}(\sec x) = \sec x \tan x$$ Integrate v'(x) with respect to x to get v(x): $$v(x) = \int \sec^2 x dx = \tan x$$

Apply the integration by parts formula

Substitute u(x), u'(x), and v(x) into the integration by parts formula: $$\int \sec^3 x dx = \int (\sec x)(\sec^2 x) dx = \int u(x)v'(x) dx = u(x)v(x) - \int u'(x)v(x) dx$$ Plugging in the computed values, we get: $$\int \sec^3 x dx = (\sec x)(\tan x) - \int (\sec x \tan x)(\tan x) dx$$

Simplify the integral

Now simplify the integral on the right side of the equation: $$\int \sec^3 x dx = \sec x \tan x - \int \sec x \tan^2 x dx$$ Recall that $$\tan^2 x = \sec^2 x - 1$$, so substituting that expression into the integral, we get: $$\int \sec^3 x dx = \sec x \tan x - \int \sec x (\sec^2 x - 1) dx$$ Now split the integral into two parts: $$\int \sec^3 x dx = \sec x \tan x - \int \sec^3 x dx + \int \sec x dx$$

Solve for the original integral

Notice that we have the original integral ($$\int \sec^3 x dx$$) on both sides of the equation. Add it to both sides to isolate the original integral: $$2\int \sec^3 x dx = \sec x \tan x + \int \sec x dx$$ Now, divide both sides by 2 to get the final result: $$\int \sec^3 x dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \int \sec x dx$$ This is the same expression we wanted to show, proving that the given statement is true.

Key concepts

Definite Integrals

Definite integrals are a vital concept in calculus, representing the signed area under a curve between two specific points on the x-axis. Unlike indefinite integrals, definite integrals have boundaries or limits of integration, which define the interval over which the integration is performed. These limits are usually denoted as \(a\) and \(b\), forming the integral \( \int_{a}^{b} f(x) \, dx \).

When evaluating a definite integral, you essentially calculate the difference in the antiderivative's values at these two bounds. This is executed through the Fundamental Theorem of Calculus, which connects differentiation and integration. The theorem states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then:

\[ \int_{a}^{b} f(x) \; dx = F(b) - F(a) \]

This process provides a numerical value, equating to the net area beneath the curve, subtracting any area below the x-axis.

• This can involve geometric interpretations such as rectangles and trapezoids to approximate areas when the function's form is complex.
• It helps in calculating physical properties such as displacement and total accumulated change over an interval.

Understanding definite integrals allows for an appreciation of how calculus links aspects of continuous rates of change with tangible quantities.

Indefinite Integrals

The concept of indefinite integrals centers around finding the antiderivative of a function. Unlike definite integrals, indefinite integrals do not have upper or lower limits, and therefore the result is a function rather than a number. This integral is represented as \( \int f(x) \, dx \) and includes a constant of integration, often denoted as \( C \), because antiderivatives are not unique.

The process of finding an indefinite integral, or the antiderivative, involves understanding what function, when differentiated, will yield the original function \( f(x) \). This is essentially the reverse of differentiation.

For example, given the function \( f(x) = x^2 \), the indefinite integral is:

\[ \int x^2 \, dx = \frac{1}{3}x^3 + C \]

• Indefinite integrals are fundamental in solving differential equations, leading to functions that describe physical phenomena.
• They underpin the derivation of formulas in field like physics, where continuous change needs tracking.

The constant \( C \) is crucial because it accounts for all possible horizontal shifts of the antiderivative graph. Thus, understanding indefinite integrals broadens insight into the wide versatility in solving integration tasks.

Trigonometric Integration

Trigonometric integration is a specialized technique for integrating functions that involve trigonometric identities. This method becomes particularly useful when dealing with integrals involving powers of sine, cosine, tangent, and their related functions.

Many trigonometrical integrals require the use of strategies such as substitution or trigonometric identities to simplify the problem. The provided example \( \int \sec^3 x \, dx \) illustrates the necessity to manipulate these identities for simplification and accurate integration.

For example:

• Substitutions like \( \sin^2 x + \cos^2 x = 1 \) or other identities can transform the integral into a more manageable form.
• Integration by parts, used in the exercise, is a technique often combined with trigonometric identities to decompose the integral into simpler parts.

The principle of trigonometric integration aids in solving complex integrals that appear frequently in areas such as physics and engineering, where wave functions and oscillating curves are common. Through understanding and applying these identities, one can tackle integrals that might initially seem daunting, efficiently finding solutions to problems that incorporate these fundamental mathematical functions.