Other Integration Concepts and Formulas in Calculus

Other Integration Concepts and Formulas

2.1 Inverse Trigonometric Functions

Integrals that result in inverse trigonometric functions are important in calculus. These integrals often arise when the integrand matches a specific form, and their evaluation requires careful attention to the domains and restrictions of the inverse trigonometric functions.

• Key Integration Formulas:

  • $\int \frac{du}{\sqrt{a^2-u^2}} = \sin^{-1}\left(\frac{u}{a}\right) + C$

  • $\int \frac{du}{a^2+u^2} = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C$

  • $\int \frac{du}{u\sqrt{u^2-a^2}} = \frac{1}{a} \sec^{-1}\left(\frac{|u|}{a}\right) + C$

• Example 1: $\int \frac{dx}{\sqrt{9-x^2}} = \sin^{-1}\left(\frac{x}{3}\right) + C$

• Example 2: $\int \frac{dx}{\sqrt{4-9x^2}} = \frac{1}{3} \sin^{-1}\left(\frac{3x}{2}\right) + C$

• Example 3: $\int \frac{1}{1+4x^2} dx = \frac{1}{2} \tan^{-1}(2x) + C$

2.2 Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are defined using exponential functions. They have similar identities and play a significant role in integration and differentiation.

• Definitions:

  • $\sinh x = \frac{e^x - e^{-x}}{2}$

  • $\cosh x = \frac{e^x + e^{-x}}{2}$

  • $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

  • $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$

  • $\text{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$

  • $\text{csch} x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$

• Key Identities:

  • $\cosh^2 x - \sinh^2 x = 1$

  • $1 - \tanh^2 x = \text{sech}^2 x$

  • $\coth^2 x - 1 = \text{csch}^2 x$

  • $\sinh 2x = 2 \sinh x \cosh x$

  • $\cosh 2x = \cosh^2 x + \sinh^2 x$

• Differentiation and Integration:

  • $(\sinh x)' = \cosh x$

  • $(\cosh x)' = \sinh x$

  • $(\tanh x)' = \text{sech}^2 x$

  • $(\coth x)' = -\text{csch}^2 x$

  • $(\text{sech} x)' = -\text{sech} x \tanh x$

  • $(\text{csch} x)' = -\text{csch} x \coth x$

  • $\int \cosh x dx = \sinh x + C$

  • $\int \sinh x dx = \cosh x + C$

  • $\int \text{sech}^2 x dx = \tanh x + C$

  • $\int \text{csch}^2 x dx = -\coth x + C$

  • $\int \text{sech} x \tanh x dx = -\text{sech} x + C$

  • $\int \text{csch} x \coth x dx = -\text{csch} x + C$

• Example 1: $\int \frac{\cosh x}{2 + 3 \sinh x} dx = \frac{1}{3} \ln|2 + 3 \sinh x| + C$

• Example 2: $\int \sinh^2 x dx = \frac{1}{4} \sinh 2x - \frac{x}{2} + C$

• Example 3: $\int e^x \sinh x dx = \frac{e^{2x}}{4} - \frac{x}{2} + C$

2.3 General Power Formula

The general power rule of integration is a powerful tool for integrating functions raised to a power, provided the derivative of the inner function is present in the integrand.

• Formula:

  • $\int [f(x)]^n f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + C$

• Example 1: $\int \frac{1}{x^3} (1 + \frac{1}{x^2})^{5/3} dx = \frac{3}{16} (1 + \frac{1}{x^2})^{8/3} + C$

• Example 2: $\int \sin^{1/3} x \cos x dx = 3 \frac{\sin^{4/3} x}{4} + C$

2.4 Constant of Integration

The constant of integration, denoted as C, is an essential part of indefinite integrals. It accounts for the family of antiderivatives that differ by a constant. When integrating sums or differences, each term technically has its own constant, but these are usually combined into a single constant for simplicity.

• Key Points:

  • Each indefinite integral technically has its own constant of integration.

  • When integrating sums, the constants are combined into a single constant.

  • Multiplying the constant by a scalar does not change its arbitrary nature.

• Example: $\int 15x^4 - 9x^{-2} dx = 3x^5 + 9x^{-1} + C$

2.5 Definite Integrals

A definite integral computes the net area under a curve between two bounds, a and b. It is denoted as $\int_a^b f(x) dx$ and has several important properties and theorems.

• Key Properties and Theorems:

  • If $f$ is integrable on $[a, b]$, then $\int_a^b k dx = k(b-a)$ for any constant $k$.

  • Linearity: $\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx$

  • If $f(x) \geq g(x)$ on $[a, b]$, then $\int_a^b f(x) dx \geq \int_a^b g(x) dx$

  • If $m \leq f(x) \leq M$ for $x$ in $[a, b]$, then $m(b-a) \leq \int_a^b f(x) dx \leq M(b-a)$

  • Partition property: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ for $a < c < b$

• Example 1: $\int_{-3}^5 4 dx = 32$

• Example 2: $\int_1^3 (3x^2 - 5x + 2) dx = 10$

• Example 3: For $f(x) = \sqrt{\sin x}$ on $[\frac{\pi}{4}, \frac{3\pi}{4}]$, $1.32 \leq \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sqrt{\sin x} dx \leq 1.57$

References and Further Reading

• Louis Leithold. The Calculus With Analytic Geometry. Sixth Edition. pp.331-339.

• Integrals Resulting in Inverse Trigonometric Functions

• Integration of Hyperbolic Functions

• General Power Rule of Integration