Indefinite-integration Formula Sheet

Indefinite integration is the process in calculus of finding a general antiderivative (or indefinite integral) of a function, without specifying the limits of integration, typically denoted by the symbol ∫.

Neetesh Kumar | June 02, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Introduction to Indefinite Integration
• 2. Standard Results
• 3. Techniques of Integration
• 4. Various Forms of the Partial Fraction

1. Introduction to Indefinite Integration

If $f$ & $F$ are functions of $x$ such that $F' (x) = f(x)$ then the function $F$ is called a Primitive or Antiderivative or Integral of $f(x)$ w.r.t. $x$ and is written symbolically as:
$\int f(x) \, dx = F(x) + c \iff \frac{d}{dx}(F(x) + c) = f(x)$ where $c$ is called the constant of integration.

Note:

If $\int f(x) \, dx = F(x) + c$, then $\int f(ax + b) \, dx = \frac{F(ax + b)}{a} + c, \, a \ne 0$

2. Standard Results

(i) $\int (ax + b)^n \, dx = \frac{(ax + b)^{n+1}}{a(n+1)} + c; \, n \ne -1$

(ii) $\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln |ax + b| + c$

(iii) $\int e^{ax + b} \, dx = \frac{1}{a} e^{ax + b} + c$

(iv) $\int \frac{dx}{a \cos^2(ax + b)} = \frac{1}{a} \tan(ax + b) + c$

(v) $\int \sin(ax + b) \, dx = -\frac{1}{a} \cos(ax + b) + c$

(vi) $\int \cos(ax + b) \, dx = \frac{1}{a} \sin(ax + b) + c$

(vii) $\int \tan(ax + b) \, dx = -\frac{1}{a} \ln |\cos(ax + b)| + c$

(viii) $\int \cot(ax + b) \, dx = \frac{1}{a} \ln |\sin(ax + b)| + c$

(ix) $\int \sec^2(ax + b) \, dx = \frac{1}{a} \tan(ax + b) + c$

(x) $\int \csc^2(ax + b) \, dx = -\frac{1}{a} \cot(ax + b) + c$

(xi) $\int \csc(ax + b) \cot(ax + b) \, dx = -\frac{1}{a} \csc(ax + b) + c$

(xii) $\int \sec(ax + b) \tan(ax + b) \, dx = \frac{1}{a} \sec(ax + b) + c$

(xiii) $\int \sec x \, dx = \ln |\sec x + \tan x| + c$

(xiv) $\int \csc x \, dx = -\ln |\csc x + \cot x| + c$

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

(xvi) $\int \frac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1} \frac{x}{a} + c$

(xvii) $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1} \frac{x}{a} + c$

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

(xix) $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + c$

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

(xxi) $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + c$

(xxii) $\int \frac{dx}{(x^2 - a^2)^2} = \frac{x}{2a^2(x^2 - a^2)} + c$

(xxiii) $\int \frac{x \, dx}{x^2 - a^2} = \frac{1}{2} \ln |x^2 - a^2| + c$

(xxiv) $\int \frac{x \, dx}{x^2 + a^2} = \frac{1}{2} \ln (x^2 + a^2) + c$

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

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

3. Techniques of Integration

(a) Substitution or change of independent variable:

Integral $I = \int f(x) \, dx$ is changed to $\int f(g(t)) g'(t) \, dt$, by a suitable substitution $x = g(t)$ provided the latter integral is easier to integrate.

Some standard substitutions:

1. $\int f(x)^n f'(x) \, dx$ put $f(x) = t$ & proceed.

2. $\int \frac{dx}{\sqrt{a^2 - x^2}}, \int \frac{dx}{\sqrt{a^2 + x^2}}, \int \frac{dx}{\sqrt{x^2 - a^2}}$ Express $ax^2 + bx + c$ in the form of a perfect square & then apply the standard results.

3. $\int \frac{px + q}{ax^2 + bx + c} \, dx$ Express $px + q$ as $A \times$ (differential coefficient of the quadratic term of the denominator) + $B$.

4. $\int e^{f(x)}(f(x) + f'(x)) \, dx = e^{f(x)} + c$

5. $\int (f(x) + x f'(x)) \, dx = x f(x) - \int x \, dx$

6. $\int \frac{dx}{x^n (x^n + 1)}$, take $x^n$ common & put $1 + x^{-n} = t$.

7. $\int \frac{dx}{x (x^2 + a^2)}$, take $x^2$ common & put $1 + x^{-2} = t$.

8. $\int \frac{dx}{x (1 + x^n)}$, take $x^n$ common and put $1 + x^{-n} = t$.

9. $\int \frac{dx}{a+bsin^2x}$ OR $\int \frac{dx}{a+bcos^2x}$ OR $\int \frac{dx}{asin^2x + bsinxcosx + ccos^2x}$
   then multiply the Numerator and Denominator by Sec$^2$x and Put tanx = t

10. $\int \frac{dx}{a+bsinx}$ OR $\int \frac{dx}{a+bcosx}$ OR $\int \frac{dx}{a+bsinx+ccosx}$
    Convert sines & cosines into their respective tangents of half the angles, put tan$\frac{x}{2} = t$

(b) Integration by parts:

$\int u.v dx = u\int v dx - \int[\frac{du}{dx}.\int v.dx]dx$

where $u$ & $v$ are differentiable functions.

Note:

While using integration by parts, choose $u$ & $v$ such that:

(i) $\int v \, dx$ & (ii) $\int[\frac{du}{dx}. v \, dx]dx$ is simple to integrate.

This is generally obtained by keeping the order of $u$ & $v$ as per the order of the letters in ILATE, where the I-Inverse function, L-Logarithmic function, A-Algebraic function, T-Trigonometric function & E-Exponential function.

(c) Partial fraction:

A rational function is defined as the ratio of two polynomials in the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials in $x$ and $Q(x) \ne 0$. If the degree of $P(x)$ is less than that of $Q(x)$, then the rational function is called proper. Otherwise, it is called improper. The long division process can reduce the improper rational function to the proper rational functions. Thus, if $\frac{P(x)}{Q(x)}$ is improper, then $\frac{P(x)}{Q(x)} = T(x) + \frac{P_1(x)}{Q(x)}$

where $T(x)$ is a polynomial in $x$ and $\frac{P_1(x)}{Q(x)}$ is a proper rational function. Writing the integrand as a sum of simpler rational functions by partial fraction decomposition is always possible. After this, the integration can be carried out easily using the already-known methods.

4. Various Forms of the Partial Fraction

(i) $\frac{2px + qx + r}{(x - a) (x - b)(x - c)} = \frac{A}{x - a} + \frac{B}{x - b} + \frac{C}{x - c}$

(ii) $\frac{2px + qx + r}{(x - a)^2 (x - b)} = \frac{A}{x - a} + \frac{B}{(x - a)^2} + \frac{C}{x - b}$

(iii) $\frac{2px + qx + r}{(x - a)(x^2 + bx + c)} = \frac{A}{x - a} + \frac{Bx + C}{x^2 + bx + c}$

where $x^2 + bx + c$ cannot be factorized further

Note:
In competitive exams, partial fractions are generally found by inspection by noting the following fact:
$\frac{1}{(x - a)(x - b)} = \frac{1}{a - b} \left(\frac{1}{x - a} - \frac{1}{x - b}\right)$

It can be applied when $x^2$ or any other function exists in all places of $x$.

Examples:

1. $\frac{1}{(x^2 + 1)(x^2 + 3)} = \frac{1}{2} \left(\frac{1}{t + 1} - \frac{1}{t + 3}\right)$ {take $x^2 = t$}
   

2. $\frac{1}{x^4 + x^2} = \frac{1}{x^2(x^2 + 1)} = \frac{1}{2} \left(\frac{1}{x^2} - \frac{1}{x^2 + 1}\right)$

3. $\frac{x}{x^3 + x^2} = \frac{1}{x(x + 1)} = \frac{1}{x} - \frac{1}{x + 1}$

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