2. INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS

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• Order of ILATE - Watch Now
• Definite Integrals - Watch Now
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2. INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS

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2.1 Integration as Inverse Operation of Differentiation

Integration is the reverse operation of differentiation. While differentiation gives the rate of change of a function, integration finds the accumulated area under the curve of a function, which essentially "undoes" differentiation.

For example:

• If $f(x) = x^2$, then the derivative of $f(x)$ is: $$\frac{d}{dx} x^2 = 2x$$
  
• The inverse of this operation is integration: $$\int 2x \, dx = x^2 + C$$ Where $C$ is the constant of integration.

Key Formula:

$$\frac{d}{dx} \left( \int f(x) \, dx \right) = f(x)$$

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2.2 Simple Integration by Substitution, by Parts, and by Partial Fractions (for Linear Factors Only)

(a) Integration by Substitution:

Substitution is a technique used when an integral contains a composite function, i.e., a function of a function. We let:

$$u = g(x)$$

Then, the differential $du = g'(x) dx$, which helps simplify the integral.

Example:

$$\int 2x \cdot \sqrt{x^2 + 1} \, dx$$

Let $u = x^2 + 1$, so $du=2xd\mathrm{x.\; Thus,\; the\; integral\; becomes:}$

$$\int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C$$

Substitute back $u = x^2 + 1$:

$$\frac{2}{3} (x^2 + 1)^{3/2} + C$$

(b) Integration by Parts:

The formula for integration by parts is derived from the product rule of differentiation:

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

Example:

$$\int x \cdot e^x \, dx$$

Let:

• $u = x$ and $dv = e^x \, dx$
  
• Then, $du = dx$ and $v=e^x$

Using the formula:

$$\int x \cdot e^x \, dx = x \cdot e^x - \int e^x \, dx$$
$$= x \cdot e^x - e^x + C$$

(c) Integration by Partial Fractions (Linear Factors Only):

When a rational function (i.e., a fraction of two polynomials) has a denominator that factors into linear factors, we can break it into simpler fractions.

For example:

$$\int \frac{1}{(x+1)(x+2)} \, dx$$

We decompose it into:

$$\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$$

Multiply both sides by $(x+1)(x+2)\; and\; solve\; for$$A$ and $B$:

$$1 = A(x+2) + B(x+1)$$

Equating coefficients, we get $A=-\mathrm{1\; and }$$B=1.$

Thus:

$$\int \frac{1}{(x+1)(x+2)} \, dx = \int \left( \frac{-1}{x+1} + \frac{1}{x+2} \right) dx$$
$$= -\ln|x+1| + \ln|x+2| + C$$

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2.3 Use of Formulas for Solving Problems Involving $\sin^m x \cdot \cos^n x \, dx$ for Positive Integers $m$ and $n$

These integrals involve powers of sine and cosine. The common formulas for solving these types of integrals are:

• For even powers of sine or cosine:
  • Use trigonometric identities to reduce the powers of sine or cosine.
  • For example:
  $$\sin^2 x = \frac{1 - \cos(2x)}{2}, \quad \cos^2 x = \frac{1 + \cos(2x)}{2}$$

Example 1:

$$\int \sin^2 x \, dx$$

Using the identity:

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

• For odd powers of sine or cosine:
  • Use substitution by factoring out one factor of sine or cosine and applying a suitable identity.

Example 2:

$$\int \sin^3 x \cdot \cos^2 x \, dx$$

Factor out $\sin x$:

$$\int \sin^3 x \cdot \cos^2 x \, dx = \int \sin x \cdot \sin^2 x \cdot \cos^2 x \, dx$$

Use the identity ${\sin }^{2}x=1-{\cos }^2\mathrm{x:}$

$$\int \sin x \cdot (1 - \cos^2 x) \cdot \cos^2 x \, dx$$

This can be simplified and solved by substitution or further identities.

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2.4 Definition of Differential Equation, Order and Degree of Differential Equation

A Differential Equation is an equation that relates a function with its derivatives. It represents how a quantity changes with respect to another quantity.

Order of a Differential Equation:

The order of a differential equation is the highest derivative that appears in the equation.

For example:

• In the equation $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$, the highest derivative is $\frac{d^2y}{dx^2}$, so the order is 2.

Degree of a Differential Equation:

The degree of a differential equation is the power of the highest derivative after the equation has been made polynomial in derivatives (i.e., no fractional or negative powers of derivatives).

For example:

• The equation $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + y = 0$ is of order 2 and degree 2 because the highest derivative is squared.

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Summary of Key Concepts and Formulas:

1. Integration as Inverse of Differentiation:

   $$\frac{d}{dx} \left( \int f(x) \, dx \right) = f(x)$$
   

2. Integration by Substitution: Let $u = g(x)$, $du=g^{\mathrm{\prime }}(x)$.

   $$\int f(g(x))g'(x) \, dx = F(u) + C$$
   

3. Integration by Parts:

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

4. Partial Fraction Decomposition:

   $$\frac{1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}$$

5. Formulas for Integrals Involving Powers of Sine and Cosine:

   • For even powers: Use trigonometric identities.
   • For odd powers: Factor out one factor and apply substitution.

6. Differential Equations:

• Order: Highest derivative.
• Degree: Power of the highest derivative after making the equation polynomial.

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  • Order of ILATE - Watch Now
  • Definite Integrals - Watch Now
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