Indefinite Integration

1. Definition and Properties

• Definition of Integration

  Integration is finding the function $g(x)$ whose derivative is known as $f(x)$.

  If $g'(x) = f(x)$, then $\int f(x) dx = g(x) + c$, where $c$ is an arbitrary constant.

  Family of curves representing integrals $f(x)=x^2+c$
• Fundamental Theorems

  Theorem 1: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

  Theorem 2: $\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx$

  Theorem 3: $\int k[f(x)] dx = k \int f(x) dx$, where $k$ is a constant.

2. Elementary Integration Formulae

Trigonometric Functions

3. Integration by Substitution

• Theorem & Corollaries

  Theorem 1: If $x = \phi(t)$ is a differentiable function of $t$, then $\int f(x) dx = \int f[\phi(t)]\phi'(t) dt$.

  Corollary I: If $\int f(x) dx = g(x) + c$, then $\int f(ax+b) dx = g(ax+b)\frac{1}{a} + c$.

  Corollary II: $\int [f(x)]^n \cdot f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + c, \quad n \neq -1$

  Corollary III: $\int \frac{f'(x)}{f(x)} dx = \log|f(x)| + c$

  Corollary IV: $\int \frac{f'(x)}{\sqrt{f(x)}} dx = 2\sqrt{f(x)} + c$

4. Some Special Integrals

5. Standard Procedures & Substitutions

• Quadratic Denominators

  Form: $\int \frac{1}{ax^2+bx+c} dx$ \ or \ $\int \frac{1}{\sqrt{ax^2+bx+c}} dx$

  Write $ax^2+bx+c$ as $a(x^2+\frac{b}{a}x+\frac{c}{a})$. Complete the square by adding and subtracting $(\frac{1}{2} \times \text{coefficient of } x)^2$ to convert it to a special integral form.

• Linear / Quadratic Form

  Form: $\int \frac{px+q}{ax^2+bx+c} dx$ \ or \ $\int \frac{px+q}{\sqrt{ax^2+bx+c}} dx$

  Express the numerator as: $px+q = A \cdot \frac{d}{dx}(ax^2+bx+c) + B$ and solve for $A$ and $B$.

• Trigonometric Forms

  Form $\int \frac{1}{a\sin^2x+b\cos^2x+c} dx$: Divide numerator and denominator by $\cos^2x$. Replace $\sec^2x$ by $1+\tan^2x$ and substitute $\tan x = t$.

  Form $\int \frac{1}{a\sin x+b\cos x+c} dx$: Substitute $\tan(\frac{x}{2}) = t \implies dx = \frac{2}{1+t^2} dt$, \ $\sin x = \frac{2t}{1+t^2}$, \ $\cos x = \frac{1-t^2}{1+t^2}$.

6. Integration by Parts

• The Theorem

  If $u$ and $v$ are two differentiable functions of $x$:

  $$ \int u v \,dx = u \int v \,dx - \int \left(\frac{d}{dx}u \cdot \int v \,dx\right) dx $$

  LIATE Rule: Proper choice of the first function ($u$) follows the order: L (Logarithmic), I (Inverse trig), A (Algebraic), T (Trigonometric), E (Exponential).

• Derived Special Integrals

  $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + c$

  $\int \sqrt{a^2+x^2} dx = \frac{x}{2}\sqrt{a^2+x^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + c$

  $\int \sqrt{x^2-a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + c$

• Special Exponential Integral

  $$ \int e^x [f(x) + f'(x)] dx = e^x \cdot f(x) + c $$

7. Integration by Partial Fraction

If the degree of the numerator $\ge$ degree of the denominator, first divide to express it as $\text{Quotient} + \frac{\text{Remainder}}{g(x)}$. For proper rational functions, use the forms below: