Summary

Master List of Standard Indefinite Integrals

Always remember to add the constant of integration, $C$, to every indefinite integral, because indefinite integration gives a family of anti-derivatives.

If $\frac{d}{dx}[F(x)] = f(x)$, then $$\int f(x)\,dx = F(x) + C.$$

Basic Algebraic and Exponential Integrals

• $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad (n \neq -1)$$ This is the power rule for integration. It fails only for $n=-1$, because then the denominator becomes zero.
• $$\int \frac{1}{x}\,dx = \ln|x| + C$$ This is the special case excluded from the power rule.
• $$\int e^x\,dx = e^x + C$$ Since the derivative of $e^x$ is itself.
• $$\int a^x\,dx = \frac{a^x}{\ln a} + C \qquad (a>0,\ a\neq 1)$$ Because $$\frac{d}{dx}(a^x)=a^x\ln a.$$

Basic Trigonometric Integrals

• $$\int \sin x\,dx = -\cos x + C$$
• $$\int \cos x\,dx = \sin x + C$$
• $$\int \sec^2 x\,dx = \tan x + C$$
• $$\int \csc^2 x\,dx = -\cot x + C$$
• $$\int \sec x\tan x\,dx = \sec x + C$$
• $$\int \csc x\cot x\,dx = -\csc x + C$$

Derived Trigonometric Integrals

• $$\int \tan x\,dx = \ln|\sec x| + C = -\ln|\cos x| + C$$
• $$\int \cot x\,dx = \ln|\sin x| + C$$
• $$\int \sec x\,dx = \ln|\sec x + \tan x| + C$$
• $$\int \csc x\,dx = \ln|\csc x - \cot x| + C$$

These formulas should be memorized thoroughly, because many difficult-looking integrals become easy once they are simplified into one of these standard forms.

Six Special Integrals

These six formulas are extremely important. They are often used after completing the square or after reducing an algebraic expression to a standard form.

Without Square Roots

1. $$\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C \qquad (a\neq 0)$$
2. $$\int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C \qquad (a\neq 0)$$
3. $$\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C \qquad (a>0)$$

With Square Roots

4. $$\int \frac{dx}{\sqrt{x^2-a^2}} = \ln\left|x+\sqrt{x^2-a^2}\right| + C$$
5. $$\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C \qquad (a>0)$$
6. $$\int \frac{dx}{\sqrt{x^2+a^2}} = \ln\left|x+\sqrt{x^2+a^2}\right| + C \qquad (a>0)$$

Why these are important?

• Forms 1 and 2 appear after factorization or partial fractions.
• Form 3 appears whenever the denominator becomes a sum of two squares.
• Forms 4, 5, and 6 arise after completing the square inside a radical.

A major exam skill is recognizing that a given integral may not look standard at first, but after rewriting the denominator or radical carefully, it matches one of these six forms exactly.

Key Integration Techniques

To solve integrals efficiently, you must know which method to apply and why.

1. Integration by Substitution

This method is used when the integrand contains a function together with its derivative, or something proportional to its derivative. If $$I = \int f(g(x))g'(x)\,dx,$$ then put $$t = g(x), \qquad dt = g'(x)\,dx,$$ and the integral becomes $$I = \int f(t)\,dt.$$ This is the reverse of the chain rule.

2. Integration by Partial Fractions

This is used for rational functions of the form $$\frac{P(x)}{Q(x)},$$ where $P(x)$ and $Q(x)$ are polynomials.

• If the fraction is improper, first perform long division.
• Then decompose the proper fraction into simpler fractions:
• $$(x-a)(x-b) \rightarrow \frac{A}{x-a} + \frac{B}{x-b}$$
• $$(x-a)^2 \rightarrow \frac{A}{x-a} + \frac{B}{(x-a)^2}$$
• $$x^2+bx+c \rightarrow \frac{Ax+B}{x^2+bx+c}$$ for irreducible quadratic factors.

This technique reduces difficult rational integrals into logarithmic and inverse trigonometric standard forms.

3. Integration by Parts

This is based on the product rule of differentiation. The standard form is $$\int u\,dv = uv - \int v\,du.$$ In this chapter's notation, it can also be written as $$\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left(f'(x)\int g(x)\,dx\right)dx.$$ Use the ILATE rule to choose $u$:

• I = Inverse trigonometric
• L = Logarithmic
• A = Algebraic
• T = Trigonometric
• E = Exponential

The first function should usually be the one that becomes simpler after differentiation.

4. Special Exponential Form

A very important shortcut is: $$\int e^x[f(x)+f'(x)]\,dx = e^x f(x) + C.$$ This works because $$\frac{d}{dx}\left(e^x f(x)\right)=e^x f(x)+e^x f'(x)=e^x[f(x)+f'(x)].$$ In many competitive-exam problems, the integrand is first rearranged to create exactly this pattern.

Properties of Definite Integrals

A definite integral $$\int_a^b f(x)\,dx$$ gives a specific number. Geometrically, it represents the signed area between the curve and the x-axis over $[a,b]$.

P0 (Dummy Variable Property): $$\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$$ The variable of integration is only a symbol.

P1 (Reversal of Limits): $$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$ As a special case, $$\int_a^a f(x)\,dx = 0.$$

P2 (Splitting Property): $$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$$ This is especially useful for modulus functions and piecewise functions.

P3 (King's Rule): $$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$$ A very powerful symmetry property.

P4 (Special Case of King's Rule): $$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$$ This is used very often with trigonometric limits like $0$ to $\pi/2$.

P5 (Even–Odd Property for Symmetric Limits): If the limits are symmetric, i.e. $[-a,a]$, then: $$\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \quad \text{if } f(-x)=f(x) \text{ (even)}$$ $$\int_{-a}^{a} f(x)\,dx = 0 \quad \text{if } f(-x)=-f(x) \text{ (odd)}$$

P6 (Symmetry about the Midpoint in $[0,2a]$): If $$f(2a-x)=f(x),$$ then $$\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx.$$ If $$f(2a-x)=-f(x),$$ then $$\int_0^{2a} f(x)\,dx = 0.$$

These properties often allow you to evaluate definite integrals without directly integrating. In JEE Main and Advanced, this is one of the biggest time-saving ideas.

Exam Tips for Board Exams

Board exams reward clarity, method, and presentation as much as the final answer.

• Always write '+ C' in indefinite integrals. This is one of the most common avoidable mistakes.

• State the method or property used. For example, write:

• "Using substitution, let $t = 1+x^2$"

• "Using integration by parts, let $u=\log x$ and $dv=x\,dx$"

• "Using the property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$"

• Show decomposition in partial fractions clearly. Write the assumed form first, then solve for constants $A, B, C$ step by step. Examiners usually award marks for these intermediate steps.

• For integration by parts, explicitly define $u$ and $dv$. Do not jump directly to the result. State why your choice is reasonable using ILATE when appropriate.

• For modulus functions, always locate the roots first. Example: for $$\int |x^2-5x+6|\,dx,$$ first factor $$x^2-5x+6=(x-2)(x-3),$$ then determine the sign in each interval, and only after that split the integral.

• In definite integrals, do not add '+ C'. Since the answer is numerical, the constant of integration is not written.

• Keep limits and substitutions consistent. If you change variables in a definite integral, also change the limits.

Exam Tips for JEE Main & Advanced

JEE problems often depend less on long calculation and more on identifying the right trick quickly.

• King's Rule is often the first attack. For complicated definite integrals with limits such as $0$ to $\pi/2$, $0$ to $\pi$, or $1$ to $3$, immediately test whether replacing $x$ by $a+b-x$ simplifies the expression.

• Always test parity for symmetric limits. If the limits are $-a$ to $a$, check whether the integrand is odd or even before doing any work. This can instantly reduce a difficult integral to either $0$ or twice a half-interval integral.

• For expressions like $\int \frac{dx}{x^m(1+x^n)^p}$, factor out the highest power of $x$. This transforms the integrand into powers of $x^{-1}$ and often makes substitution straightforward.

• Be comfortable with the Fundamental Theorem of Calculus and Leibniz Rule. Problems involving limits and integrals frequently reduce to differentiation of an integral expression.

• Riemann sums must be recognized instantly. Any limit of the form $$\lim_{n\to\infty}\frac1n\sum_{r=1}^n f\left(\frac{r}{n}\right)$$ should immediately suggest $$\int_0^1 f(x)\,dx.$$

• Simplify before integrating. In JEE, a difficult integral is often solved not by brute force, but by rewriting the integrand cleverly using identities, algebraic manipulation, or symmetry.

• For logarithmic and trigonometric definite integrals, pairing the original integral with its transformed version is a standard trick. This often turns a complicated integrand into $1$, a constant, or a much simpler expression.

A strong JEE student does not memorize only formulas; they also recognize patterns quickly and know which transformation is most likely to work.