# UNIBEN MTH 218 Mathematical Methods I: Study Guide and Practice

> A study guide for UNIBEN MTH 218 (Mathematical Methods I): integration techniques, partial derivatives, Lagrange multipliers, Taylor series and complex numbers.

Source: https://practicecalculus.com/courses/uniben-mth-218
Updated: 2026-11-03

## At a glance

- **Course code:** MTH 218
- **Credits:** 3
- **Semester:** 200 level, first semester (core course)
- **Prerequisites:** MTH 110 and MTH 125
- **Covers:** Integration by substitution, parts and partial fractions, reduction formulae, partial differentiation and critical points of functions of two variables, Lagrange multipliers, polar, cylindrical and spherical coordinates, Taylor and Maclaurin series, nth derivatives and Leibniz's rule, complex numbers and hyperbolic functions
- **Next course:** MTH 228 Mathematical Methods II (second semester)
- **Practice here:** [Calculus 2](/learn/calculus-2) and [partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) in [Calculus 3](/learn/calculus-3)

## Key takeaways

- MTH 218 Mathematical Methods I is a 3-credit core course in the first semester of 200 level at the University of Benin.
- It starts with integration techniques: substitution, parts and partial fractions.
- It then covers partial differentiation, classifying critical points of functions of two variables and Lagrange multipliers.
- Coordinate systems, Taylor and Maclaurin series, Leibniz's rule, complex numbers and hyperbolic functions complete the course.

**UNIBEN MTH 218**, Mathematical Methods I, is the main 200-level calculus course at the University of Benin. It is a 3-credit core course in the first semester of 200 level, with MTH 110 and MTH 125 as its listed prerequisites, and it builds on the calculus of [MTH 112](/courses/uniben-mth-112). This guide maps every MTH 218 topic to free practice, so you can learn each method by solving problems with instant feedback.

## What UNIBEN MTH 218 covers

The department's description opens with techniques of integration: substitution, integration by parts and partial fractions. It continues with differentiation topics, including reduction formulae, partial differentiation with applications, and classifying the critical points of functions of two variables, followed by Lagrange multipliers.

Next come coordinate systems (changing from Cartesian to polar, spherical and cylindrical coordinates), Taylor's and Maclaurin's series, nth-order derivatives and Leibniz's rule with applications to differential equations. The course ends with complex numbers and hyperbolic functions: De Moivre's theorem, roots of complex numbers and polynomials, the exponential form, and functions of a complex variable.

In standard terms, MTH 218 combines the techniques and series of **Calculus 2** with the partial derivative and optimization part of **Calculus 3**, plus complex numbers.

## MTH 218 topic map

| MTH 218 topic | Practice | Read |
| --- | --- | --- |
| Integration by substitution | [U-substitution refresher](/learn/calculus-2/integration-techniques/u-substitution-refresher) | [U-substitution](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Reduction formulae | [Powers of sine and cosine](/learn/calculus-2/integration-techniques/trig-integrals) | [Integration by parts rule](/rules/integration-by-parts-rule) |
| Partial differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Critical points of functions of two variables | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Power series](/blog/power-series) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Taylor and Maclaurin series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| nth derivatives and Leibniz's rule | [The product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product rule](/rules/product-rule) |
| Hyperbolic functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Hyperbolic functions](/functions/hyperbolic-functions) |
| Complex numbers and De Moivre's theorem | [Key trig values](/learn/calculus-1/functions-warmup/key-trig-values) | [Natural exponential function](/functions/natural-exponential-function) |

## MTH 218 in the UNIBEN sequence

| Course | Level and semester | Credits | Focus |
| --- | --- | --- | --- |
| MTH 112 Calculus | 100, first | 3 | Limits, differentiation, integration |
| MTH 218 Mathematical Methods I | 200, first | 3 | Integration techniques, partial differentiation, series, complex numbers |
| MTH 213 Vector Analysis | 200, first | 3 | Vector products, space curves, grad, div and curl |
| MTH 228 Mathematical Methods II | 200, second | 3 | Differential equations, Fourier series, PDEs |

MTH 218 and [MTH 213 Vector Analysis](/courses/uniben-mth-213) run in the same semester, and together they cover most of a standard Calculus 3 course. If your 100-level calculus is rusty, start with the [MTH 112 guide](/courses/uniben-mth-112).

## Example problems at the MTH 218 level

**1. Partial fractions.** Since $\dfrac{1}{x^2 - 1} = \dfrac{1}{2}\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1}\right)$,

$$\int \frac{dx}{x^2 - 1} = \frac{1}{2}\ln\left|\frac{x - 1}{x + 1}\right| + C$$

**2. Classifying a critical point.** For $f(x, y) = x^2 + y^2 - 2x + 4y$, set $f_x = 2x - 2 = 0$ and $f_y = 2y + 4 = 0$, giving $(1, -2)$. With $f_{xx} = 2$, $f_{yy} = 2$ and $f_{xy} = 0$,

$$D = f_{xx}f_{yy} - f_{xy}^2 = 4 > 0, \quad f_{xx} > 0$$

so $(1, -2)$ is a minimum, with $f(1, -2) = -5$.

**3. Changing coordinates.** The Cartesian point $(1, \sqrt{3})$ has $r = \sqrt{1 + 3} = 2$ and $\tan\theta = \sqrt{3}$, so in polar form it is $(2, \frac{\pi}{3})$.

**4. Roots of unity with De Moivre.** The solutions of $z^3 = 1$ are $z = \cos\frac{2k\pi}{3} + i\sin\frac{2k\pi}{3}$ for $k = 0, 1, 2$:

$$z = 1, \quad -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad -\frac{1}{2} - \frac{\sqrt{3}}{2}i$$

## How to do well in UNIBEN MTH 218

- **Choose the integration method before you start.** Ask whether the integrand suits substitution, parts or partial fractions. Our [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson trains exactly this decision.
- **Learn the second-derivative test for two variables.** Most critical point questions come down to computing $D = f_{xx}f_{yy} - f_{xy}^2$ correctly.
- **Memorize key series.** The Maclaurin series for $e^x$, $\sin x$, $\cos x$ and $\ln(1 + x)$ are on the [calculus formula sheet](/formulas) and come up repeatedly.
- **Watch your algebra.** Partial fraction decompositions and complex roots are full of sign traps; see [common calculus mistakes](/blog/common-calculus-mistakes).
- **Practice with timed sets** from [practice exams](/practice) before each test.

> **Tip:** hyperbolic functions behave like trig functions with small sign changes: $\cosh^2 x - \sinh^2 x = 1$ instead of $\cos^2 x + \sin^2 x = 1$. Learn the definitions in terms of $e^x$ and every identity follows.

## Frequently asked questions

### What is MTH 218 at UNIBEN?

MTH 218 Mathematical Methods I is a 3-credit, first-semester, 200-level core course in the University of Benin Department of Mathematics. It covers integration techniques, partial differentiation and critical points, Lagrange multipliers, coordinate systems, Taylor and Maclaurin series, nth derivatives with Leibniz's rule, complex numbers and hyperbolic functions.

### What are the prerequisites for MTH 218?

The department lists MTH 110 Algebra and Trigonometry and MTH 125 Differential Equations and Dynamics. The course also relies on the differentiation and integration taught in MTH 112 Calculus.

### Is MTH 218 the same as Calculus 2 or Calculus 3?

It mixes both. Integration techniques and Taylor series are Calculus 2 topics, while partial derivatives, critical points of two-variable functions, Lagrange multipliers and cylindrical and spherical coordinates are Calculus 3 topics.

### What comes after MTH 218?

MTH 228 Mathematical Methods II follows in the second semester of 200 level, with differential equations, Fourier series and partial differential equations.

### How should I prepare for MTH 218?

Revise differentiation and integration from MTH 112 and complex numbers from MTH 110. Then practice partial fractions and integration by parts until they are routine, since the course opens with them.