---
title: "Evaluate ∫(x²+iy)dz along y = x"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:complex-integral-path-y-x"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-path-y-x
site: BCT Engineering
---

# Evaluate ∫(x²+iy)dz along y = x

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2070 Magh, 5 marks
- Video: [Evaluate the integration limit (0, 1+i) ∫ (x^2-iy) dz, along the path 1) y=x, 2) y=x^2](https://www.youtube.com/watch?v=VZN01jXjfWk) — Engineering & 12(Sci)  with Amey Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-path-y-x
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Evaluate ∫₀^{1+i} (x² + iy) dz along the path y = x.

## Short answer

Parametrise path y=x: z=(1+i)x, dz=(1+i)dx, x∈[0,1], and f = x^2+iy = x^2+ix. Integral = (1+i)∫_0^1 (x^2+ix)dx = (1+i)(1/3 + i/2) = −1/6 + (5/6)i.

## Answer

#### Complex integral
If f(z) is continuous on a curve C, the **complex integral** is

∫_C f(z) dz = ∫_C (u dx − v dy) + i ∫_C (v dx + u dy),

or ∫ f(z(t)) z′(t) dt for a parameterisation z = z(t).

#### Solution
f = x² + iy, path y = x from z = 0 to z = 1 + i.

Parameterise: z = x + ix = (1 + i)x, dz = (1 + i) dx, with x from 0 to 1. On the path, x² + iy = x² + ix.

```text
I = ∫₀¹ (x² + i x)(1 + i) dx
  = (1 + i) [ x³/3 + i x²/2 ]₀¹
  = (1 + i)(1/3 + i/2)
  = 1/3 + i/2 + i/3 − 1/2
  = −1/6 + 5i/6
```

**Answer: −1/6 + (5/6) i**

(The value happens to coincide with the result along y = x². This is not general, because x² + iy is not analytic, so the integral is normally path dependent.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. On y = x, dz =
   A. (1+i)dx
   B. dx
   C. (1+2ix)dx
   D. i dx
   Answer: A. z = (1+i)x. (Key point: Parametrise)

2. On the path x² + iy =
   A. (1+i)x
   B. x²
   C. x² + ix
   D. x² + ix²
   Answer: C. y=x. (Key point: Integrand)

3. ∫₀¹(x²+ix)dx =
   A. 1/3 + i
   B. 1 + i
   C. 1/3 + i/2
   D. 1/2 + i/3
   Answer: C. x³/3 + ix²/2. (Key point: Antiderivative)

4. Value is the same as along y=x² because:
   A. Both paths same
   B. f is analytic
   C. Always independent
   D. It is coincidence; x²+iy is not analytic so generally path dependent
   Answer: D. Not analytic. (Key point: Path dependence)

## Related in Applied Mathematics

- [Evaluate ∫(x²+iy)dz along y = x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x2) — Applied Mathematics, chapter 1
- [Define complex integral; evaluate ∮log z dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-log-z) — Applied Mathematics, chapter 1
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