---
title: "Evaluate ∫∫ e^(y/x) dy dx"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:evaluate-e-y-x-dy"
source: https://www.bctengineer.com/q/semester_2/mathematics/evaluate-e-y-x-dy
site: BCT Engineering
---

# Evaluate ∫∫ e^(y/x) dy dx

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2071 Bhadra, 5 marks
- Video: [DOUBLE INTEGRAL Evaluate ∫∫ e^(y/x)dydx](https://www.youtube.com/watch?v=MDoiE6CG-FQ) — m-easy maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/evaluate-e-y-x-dy
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∫₀¹ ∫₀^(x²) e^(y/x) dy dx.

## Short answer

Integrate inner: e^(y/x) dy from 0 to x² gives x eˣ − x. Then ∫₀¹ (x eˣ − x) dx = 1 − 1/2 = 1/2.

## Answer

I = ∫₀¹ ∫₀^{x²} e^(y/x) dy dx

The integral is easy in the given order, because integrating e^(y/x) with respect to y just brings out a factor x.

#### Inner integral (x constant)
```text
∫₀^{x²} e^(y/x) dy = [ x e^(y/x) ]₀^{x²}
                   = x e^x − x
```

#### Outer integral
```text
I = ∫₀¹ (x eˣ − x) dx
  = ∫₀¹ x eˣ dx − ∫₀¹ x dx
```

By parts: ∫ x eˣ dx = x eˣ − eˣ, so ∫₀¹ x eˣ dx = (e − e) − (0 − 1) = 1.

```text
I = 1 − 1/2 = 1/2
```

**Answer: 1/2.**

## Check yourself

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

1. ∫₀^{x²} e^(y/x) dy:
   A. x eˣ − x
   B. x² eˣ
   C. e^x
   D. eˣ − 1
   Answer: A. x[e^(y/x)]. (Key point: Inner)

2. ∫₀¹ x eˣ dx:
   A. 0
   B. 1
   C. e
   D. e−1
   Answer: B. [xeˣ−eˣ]. (Key point: By parts)

3. Value:
   A. 1/2
   B. e−1
   C. 1
   D. e/2
   Answer: A. 1 − 1/2. (Key point: Result)

## Related in Mathematics

- [Evaluate ∫∫ e^(x²) dx dy](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-x2-dx-dy) — Mathematics, chapter 2
- [Polar coordinates: ∫∫(x²+y²) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-x2-y2-dy) — Mathematics, chapter 2, asked 5×

## Related in other subjects

- [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6×
- [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6×