Advanced Techniques of Integration: Trigonometric, Inverse Trigonometric, and Exponential Integrals

Using Substitution to Evaluate Trigonometric Integrals

Substitution and Trigonometric Identities

Substitution and trigonometric identities are powerful tools for evaluating integrals involving trigonometric functions. These methods simplify complex expressions and make integration possible.

• Key Identity: $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$

• Example: To evaluate $\int_0^{\frac{\pi}{2}} \cos^2 \theta \, d\theta$, substitute the identity:

$\int_0^{\frac{\pi}{2}} \cos^2 \theta \, d\theta = \int_0^{\frac{\pi}{2}} \frac{1 + \cos 2\theta}{2} \, d\theta$

• Split the integral:

$= \frac{1}{2} \int_0^{\frac{\pi}{2}} d\theta + \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos 2\theta \, d\theta$

• Use substitution $u = 2\theta$ for the second term:

$\int_0^{\frac{\pi}{2}} \cos 2\theta \, d\theta = \frac{1}{2} \int_0^{\pi} \cos u \, du = \frac{1}{2} [\sin u]_0^{\pi} = 0$

• Final result:

$\int_0^{\frac{\pi}{2}} \cos^2 \theta \, d\theta = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}$

• Practice: Similarly, compute $\int_0^{\frac{\pi}{2}} \sin^2 x \, dx$ using $\sin^2 x = \frac{1 - \cos 2x}{2}$.

Integrals Resulting in Inverse Trigonometric Functions

Standard Forms and Applications

Certain integrals yield inverse trigonometric functions as their antiderivatives. Recognizing these forms is essential for solving a wide range of problems.

• Key Formulas:

Integral	Result
$\int \frac{du}{\sqrt{a^2 - u^2}}$	$\sin^{-1}\left(\frac{u}{|a|}\right) + C$
$\int \frac{du}{a^2 + u^2}$	$\frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C$

• Derivation Example: For $y = \sin^{-1}\left(\frac{x}{a}\right)$, implicit differentiation gives:

$\frac{dy}{dx} = \frac{1}{\sqrt{a^2 - x^2}}$

• Thus, $y = \int \frac{1}{\sqrt{a^2 - x^2}} dx$

• Definite Integral Example:

$\frac{1}{2} \int_0^1 \frac{dx}{\sqrt{1 - x^2}} = \frac{1}{2} \left[ \sin^{-1} x \right]_0^1 = \frac{\pi}{6}$

• Substitution Example: To evaluate $\int \frac{dx}{\sqrt{4 - 9x^2}}$, let $u = 3x$, $du = 3dx$:

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

• Practice: Find the antiderivative of $\int \frac{dx}{1 + 4x^2}$ by letting $u = 2x$:

$\int \frac{dx}{1 + 4x^2} = \frac{1}{2} \int \frac{du}{1 + u^2} = \frac{1}{2} \tan^{-1}(u) + C = \frac{1}{2} \tan^{-1}(2x) + C$

Integration by Parts

Formula and Application

Integration by parts is a technique based on the product rule for differentiation. It is used to integrate products of functions.

• Formula:

$\int u \, dv = uv - \int v \, du$

• Example: Evaluate $\int x \sin x \, dx$:

Let $u = x$, $dv = \sin x \, dx$ $du = dx$, $v = -\cos x$ $\int x \sin x \, dx = -x \cos x + \int \cos x \, dx = -x \cos x + \sin x + C$

Integration by Parts - Table Method

Systematic Approach for Repeated Integration by Parts

The table method streamlines repeated applications of integration by parts, especially for polynomials multiplied by exponentials or trigonometric functions.

• Table Construction: List derivatives of $u(x)$ and integrals of $v(x)$ in columns, pairing them diagonally.

Derivatives of $u(x)$	Integrals of $v(x)$
$u(x)$	$v_1(x)$
$u'(x)$	$v_2(x)$
$u''(x)$	$v_3(x)$
...	...

• General Formula:

$\int u \, dv = u v_1 - u' v_2 + u'' v_3 - \cdots$

• Example: Evaluate $\int x^3 \cos 2x \, dx$:

Construct the table for derivatives of $x^3$ and integrals of $\cos 2x$: $\int x^3 \cos 2x \, dx = x^3 \sin 2x / 2 - 3x^2 \cos 2x / 4 + 6x \sin 2x / 8 - 6 \cos 2x / 8 + C$

• Example: $\int (x^2 + 2x^2 - x + 3) \sin(3x + 4) \, dx$ (see table in notes for full expansion).

• Example: $\int x^2 e^{-2x} \, dx$ (see table in notes for full expansion).

The Product of an Exponential and a Sine or Cosine

General Solution for $\int e^{ax} \sin(bx) \, dx$ and $\int e^{ax} \cos(bx) \, dx$

Integrals involving the product of an exponential and a sine or cosine function can be solved using integration by parts and algebraic manipulation.

• General Form: $\int e^{ax} \sin(bx) \, dx$

• Let $u = e^{ax}$, $dv = \sin(bx) \, dx$

• After two applications of integration by parts, combine results to solve for the original integral.

$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C$

• Similarly: $\int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos(bx) + b \sin(bx)) + C$

• Example: Evaluate $\int e^{2x} \sin(3x) \, dx$:

$\int e^{2x} \sin(3x) \, dx = \frac{e^{2x}}{13} (2 \sin(3x) - 3 \cos(3x)) + C$

*Additional info: The notes cover advanced integration techniques relevant to Calculus II, including substitution, integration by parts, table method, and integrals resulting in inverse trigonometric functions. All examples and formulas are standard in college calculus courses.*