IIT-JEE MAINS 2027 formula sheet for Calculus! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Calculus Formula Sheet & Chapter Summary | IIT-JEE Main 2027

Calculus forms the backbone of the IIT-JEE Main Mathematics syllabus, accounting for a significant portion of the total weightage. Mastering fundamental limits, differentiation techniques, integration shortcuts, and differential equations is essential for speed and accuracy during the exam.

1. Limits, Continuity, and Differentiability

Understanding indeterminate forms is vital for solving limit problems quickly.

• Standard Limits: \(\lim_{x\to 0}\frac{\sin x}{x} = 1\), \(\lim_{x\to 0}\frac{e^x - 1}{x} = 1\), \(\lim_{x\to 0}\frac{\ln(1+x)}{x} = 1\)
• Indeterminate Form \(1^\infty\): If \(\lim_{x\to a} f(x) = 1\) and \(\lim_{x\to a} g(x) = \infty\), then \(\lim_{x\to a} [f(x)]^{g(x)} = e^{\lim_{x\to a} g(x)[f(x)-1]}\)
• L'Hôpital's Rule: Applicable for \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) forms by differentiating numerator and denominator independently.

2. Differential Calculus & Applications

Differential calculus focuses on rates of change, slopes of tangents, and function monotonicity.

• Product & Quotient Rules: \(\frac{d}{dx}(uv) = u'v + uv'\) and \(\frac{d}{dx}\bigl(\frac{u}{v}\bigr) = \frac{u'v - uv'}{v^2}\)
• Tangents & Normals: Slope of tangent at \((x_1, y_1)\) is \(m = \frac{dy}{dx}\Big|_{(x_1, y_1)}\). Equation of normal is \(y - y_1 = -\frac{1}{m}(x - x_1)\).
• Monotonicity & Extrema: A function is increasing if \(f'(x) > 0\) and decreasing if \(f'(x) < 0\). Local extrema occur where \(f'(x) = 0\) and change sign.

3. Integral Calculus & Differential Equations

Integration evaluates areas under curves and solves dynamic physical models.

• Standard Integrals: \[\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}\bigl(\frac{x}{a}\bigr) + C\] \[\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\bigl(\frac{x}{a}\bigr) + C\]
• Definite Integral Property: \(\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\)
• Linear Differential Equations: For \(\frac{dy}{dx} + P(x)y = Q(x)\), Integrating Factor \(I.F. = e^{\int P dx}\), solution is \(y \cdot (I.F.) = \int Q \cdot (I.F.) dx + C\).