Techniques of Integration: Substitution and Trigonometric Integrals

Techniques of Integration

Substitution Method

The substitution method, also known as u-substitution, is a fundamental technique for evaluating integrals, especially when the integrand is a composite function. This method simplifies integration by changing variables, making the integral easier to solve.

• Definition: If $u = g(x)$ is a differentiable function whose range is an interval $I$, and $f$ is continuous on $I$, then:

$\int f(g(x))g'(x)\,dx = \int f(u)\,du$

• Purpose: To transform a complicated integral into a simpler one by substituting part of the integrand with a new variable.

• When to Use: When the integrand contains a function and its derivative, or can be manipulated into such a form.

• Steps:

  1. Identify a substitution $u = g(x)$ that simplifies the integrand.

  2. Compute $du = g'(x)dx$.

  3. Rewrite the integral in terms of $u$ and $du$.

  4. Integrate with respect to $u$.

  5. Substitute back $x$ in terms of $u$ if required.

• Example: Evaluate $\int x \cos(x^2) dx$.

  • Let $u = x^2 \implies du = 2x dx \implies \frac{du}{2} = x dx$

  • Substitute: $\int x \cos(x^2) dx = \int \cos(u) \frac{du}{2} = \frac{1}{2} \int \cos(u) du = \frac{1}{2} \sin(u) + C = \frac{1}{2} \sin(x^2) + C$

Trigonometric Integrals

Integrals involving trigonometric functions often benefit from substitution, trigonometric identities, or both. Common strategies include using identities to simplify the integrand before integrating.

• Common Integrals:

  • $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C$

  • $\int \cos(ax) dx = \frac{1}{a} \sin(ax) + C$

  • $\int \sec^2(x) dx = \tan(x) + C$

  • $\int \csc^2(x) dx = -\cot(x) + C$

• Example: Evaluate $\int \cos^2(x) dx$.

  • Use the power-reduction identity: $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

  • $\int \cos^2(x) dx = \int \frac{1}{2} dx + \int \frac{\cos(2x)}{2} dx = \frac{x}{2} + \frac{1}{4} \sin(2x) + C$

Indefinite and Definite Integrals with Substitution

Substitution can be used for both indefinite and definite integrals. For definite integrals, the limits of integration must be changed to match the new variable.

• Changing Limits: If $x = a$ corresponds to $u = g(a)$ and $x = b$ to $u = g(b)$, then:

$\int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$

• Example: Evaluate $\int_0^1 2x e^{x^2} dx$.

  • Let $u = x^2 \implies du = 2x dx$

  • When $x = 0, u = 0$; when $x = 1, u = 1$

  • $\int_0^1 2x e^{x^2} dx = \int_0^1 e^u du = e^u \big|_0^1 = e^1 - e^0 = e - 1$

Sample Problems from Section 1.5 Exercises

• Conceptual Questions:

  • Why is u-substitution referred to as a change of variable?

  • When reversing the chain rule, what should you take as $u$?

• Verification by Differentiation: Several exercises ask to verify an antiderivative by differentiating and to use the indicated substitution to show the integral takes the given form.

• Indefinite Integrals: Many exercises require finding the antiderivative using substitution, e.g.:

  • $\int (x-1)^3 dx$ with $u = x-1$

  • $\int \cos^2(\theta) \sin(\theta) d\theta$ with $u = \cos(\theta)$

• Definite Integrals: Problems involve changing limits and evaluating, e.g.:

  • $\int_0^1 2x e^{x^2} dx$

  • $\int_1^2 \frac{1}{\sqrt{x}} dx$

• Trigonometric Substitution: Some exercises involve integrals like $\int \sin^2(x) \cos(x) dx$ or $\int \cos^3(\theta) d\theta$.

• Graphical Applications: Problems ask for estimation of area under a curve using Riemann sums and substitution for exact values.

• Applied Problems: Exercises include expressing the area of a semicircle or ellipse in terms of an integral using substitution.

Table: Common Substitutions and Their Applications

Original Integral	Substitution	Resulting Integral
$\int f(ax + b) dx$	$u = ax + b$	$\frac{1}{a} \int f(u) du$
$\int x e^{x^2} dx$	$u = x^2$	$\frac{1}{2} \int e^u du$
$\int \sin^n(x) \cos(x) dx$	$u = \sin(x)$	$\int u^n du$
$\int \cos^n(x) \sin(x) dx$	$u = \cos(x)$	$-\int u^n du$

Additional info:

• Some exercises involve graphical estimation and applications to geometry (e.g., area under curves, semicircles, ellipses) using substitution.

• Students are expected to understand both the mechanics of substitution and its application to definite and indefinite integrals, including trigonometric and exponential functions.