Engineering Mathematics Formula Sheet for All Semesters (BTech Cheatsheet)

Introduction

Engineering Mathematics is the backbone of every BTech program. From your very first semester to the final year, mathematics plays a crucial role in subjects like electronics, computer science, mechanical engineering, and civil engineering.

But let’s be honest: remembering all the formulas can feel overwhelming, especially during exams. That’s why we’ve created this Engineering Mathematics Formula Sheet for All Semesters—a concise yet comprehensive collection of formulas that covers calculus, algebra, probability, numerical methods, differential equations, and more.

This cheatsheet will help you revise quickly, save time, and score better in exams. Whether you’re in your 1st semester or preparing for final-year subjects, this formula sheet will be your go-to resource.

Why You Need an Engineering Mathematics Formula Sheet

Quick revision before exams
Easy recall of complex formulas
Covers multiple semesters in one place
Saves time during last-minute preparation
Helps in solving problems faster

Semester-Wise Engineering Mathematics Formula Sheet

We’ve broken down the formulas semester-wise so you can directly jump to the section relevant to you.

1st Semester Engineering Mathematics Formulas

Key Topics: Calculus, Trigonometry, Matrices, Algebra

Calculus Formulas

Derivative of xn=nxn−1x^n = n x^{n-1}xn=nxn−1
ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos xdxd(sinx)=cosx
ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin xdxd(cosx)=−sinx
∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1+C
∫exdx=ex+C\int e^x dx = e^x + C∫exdx=ex+C
∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C∫x1dx=ln∣x∣+C

Trigonometric Formulas

sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1sin2x+cos2x=1
sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos xsin2x=2sinxcosx
cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x – \sin^2 xcos2x=cos2x−sin2x
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}tan(A±B)=1∓tanAtanBtanA±tanB

Matrix Formulas

Determinant of 2×2: ∣A∣=ad−bc|A| = ad – bc∣A∣=ad−bc
Inverse of 2×2: A−1=1ad−bc[d−b−ca]A^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}A−1=ad−bc1[d−c−ba]
Eigenvalue equation: ∣A−λI∣=0|A – \lambda I| = 0∣A−λI∣=0

2nd Semester Engineering Mathematics Formulas

Key Topics: Differential Calculus, Vector Algebra, Multiple Integrals

Differential Calculus

dydx=lim⁡h→0f(x+h)−f(x)h\frac{dy}{dx} = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}dxdy=limh→0hf(x+h)−f(x)
Rolle’s theorem and Mean Value theorem applications
Taylor’s Series: f(x)=f(a)+f’(a)(x−a)+f’’(a)2!(x−a)2+…f(x) = f(a) + f’(a)(x-a) + \frac{f’’(a)}{2!}(x-a)^2 + …f(x)=f(a)+f’(a)(x−a)+2!f’’(a)(x−a)2+…

Vector Algebra

Gradient: ∇f=∂f∂xi+∂f∂yj+∂f∂zk\nabla f = \frac{\partial f}{\partial x}i + \frac{\partial f}{\partial y}j + \frac{\partial f}{\partial z}k∇f=∂x∂fi+∂y∂fj+∂z∂fk
Divergence: ∇⋅A\nabla \cdot A∇⋅A
Curl: ∇×A\nabla \times A∇×A

Multiple Integrals

∬Rf(x,y)dxdy\iint\limits_R f(x,y) dxdyR∬f(x,y)dxdy
Change to polar coordinates: dxdy=rdrdθdxdy = rdrd\thetadxdy=rdrdθ

3rd Semester Engineering Mathematics Formulas

Key Topics: Ordinary Differential Equations (ODEs), Laplace Transform, Fourier Series

Differential Equations

General solution: y=yc+ypy = y_c + y_py=yc+yp
First-order linear DE: dydx+Py=Q ⟹ ye∫Pdx=∫Qe∫Pdxdx+C\frac{dy}{dx} + Py = Q \implies y e^{\int P dx} = \int Q e^{\int P dx} dx + Cdxdy+Py=Q⟹ye∫Pdx=∫Qe∫Pdxdx+C

Laplace Transform

L{f(t)}=∫0∞e−stf(t)dtL\{f(t)\} = \int_0^\infty e^{-st} f(t) dtL{f(t)}=∫0∞e−stf(t)dt
L{1}=1sL\{1\} = \frac{1}{s}L{1}=s1
L{tn}=n!sn+1L\{t^n\} = \frac{n!}{s^{n+1}}L{tn}=sn+1n!
L{sin⁡at}=as2+a2L\{\sin at\} = \frac{a}{s^2+a^2}L{sinat}=s2+a2a

Fourier Series

f(x)=a02+∑n=1∞(ancos⁡nx+bnsin⁡nx)f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)f(x)=2a0+∑n=1∞(ancosnx+bnsinnx)
Coefficients:
- an=1π∫−ππf(x)cos⁡(nx)dxa_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx) dxan=π1∫−ππf(x)cos(nx)dx
- bn=1π∫−ππf(x)sin⁡(nx)dxb_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx) dxbn=π1∫−ππf(x)sin(nx)dx

4th Semester Engineering Mathematics Formulas

Key Topics: Partial Differential Equations (PDEs), Complex Analysis, Probability

PDEs

Standard form: ∂2z∂x2+∂2z∂y2=0\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0∂x2∂2z+∂y2∂2z=0 (Laplace equation)
Wave equation: ∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}∂t2∂2u=c2∂x2∂2u

Complex Analysis

Euler’s formula: eix=cos⁡x+isin⁡xe^{ix} = \cos x + i \sin xeix=cosx+isinx
Cauchy-Riemann equations: ∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}∂x∂u=∂y∂v,∂y∂u=−∂x∂v
Residue theorem: ∮f(z)dz=2πi∑Res[f(z)]\oint f(z)dz = 2\pi i \sum Res[f(z)]∮f(z)dz=2πi∑Res[f(z)]

Probability

Mean: E(X)=∑xP(x)E(X) = \sum xP(x)E(X)=∑xP(x)
Variance: Var(X)=E(X2)−[E(X)]2Var(X) = E(X^2) – [E(X)]^2Var(X)=E(X2)−[E(X)]2
Bayes’ theorem: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}P(A∣B)=P(B)P(A∩B)

5th Semester Onwards: Advanced Engineering Mathematics Formulas

Key Topics: Numerical Methods, Transform Techniques, Optimization

Numerical Methods

Newton-Raphson: xn+1=xn−f(xn)f’(xn)x_{n+1} = x_n – \frac{f(x_n)}{f’(x_n)}xn+1=xn−f’(xn)f(xn)
Trapezoidal Rule: ∫abf(x)dx≈h2[y0+2(y1+…+yn−1)+yn]\int_a^b f(x) dx \approx \frac{h}{2}[y_0 + 2(y_1+…+y_{n-1}) + y_n]∫abf(x)dx≈2h[y0+2(y1+…+yn−1)+yn]
Simpson’s 1/3 Rule: ∫abf(x)dx≈h3[y0+4(y1+y3+…)+2(y2+y4+…)+yn]\int_a^b f(x) dx \approx \frac{h}{3}[y_0 + 4(y_1+y_3+…) + 2(y_2+y_4+…) + y_n]∫abf(x)dx≈3h[y0+4(y1+y3+…)+2(y2+y4+…)+yn]

Optimization

Maxima/Minima conditions using derivatives
Lagrange multipliers: F(x,y,λ)=f(x,y)+λ(g(x,y)−c)F(x,y,λ) = f(x,y) + λ(g(x,y)-c)F(x,y,λ)=f(x,y)+λ(g(x,y)−c)

Downloadable PDF Formula Sheet

Many students prefer having a single PDF they can print and keep handy. That’s why we’ll soon provide a free downloadable PDF formula sheet covering all semesters.

FAQs on Engineering Mathematics Formula Sheet

Q1. Is this formula sheet useful for all branches of engineering?
Yes, the formulas listed here are common across all branches—CSE, ECE, Mechanical, Civil, IT, etc.—as engineering mathematics is a core subject.

Q2. Can I rely only on this sheet for exams?
This formula sheet is meant for quick revision. You should practice problems along with memorizing formulas for best results.

Q3. Will this cover GATE exam mathematics as well?
Most of the formulas here are also part of GATE syllabus. However, for GATE, practice more application-based problems.

Q4. How can I remember so many formulas easily?
Make short notes, group similar formulas together, and revise daily. Using visual aids like charts or flashcards also helps.

Q5. Is there a downloadable PDF version?
Yes, we are providing a free PDF version of the entire formula sheet for easy offline use.

Conclusion

Engineering Mathematics can seem intimidating, but with the right formula sheet, revision becomes simple and stress-free. This BTech Engineering Mathematics Formula Sheet for All Semesters is designed to save you time, boost your confidence, and help you score higher in exams.

Bookmark this page, share it with your classmates, and make it your go-to resource during exams.

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Engineering Mathematics Formula Sheet for All Semesters

Introduction

Why You Need an Engineering Mathematics Formula Sheet

Semester-Wise Engineering Mathematics Formula Sheet