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# AS-Level Maths | Integration
- URL: https://www.a-level-maths-tutor.com/as-level-maths-integration/
- Published: 2026-07-23T15:40:30.000Z
- Updated: 2026-07-23T15:40:30.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

To integrate: raise the power by one, then divide the coefficient by the new power.

$\\displaystyle\\int ax^n \\, dx = \\frac{a}{n+1}x^{n+1} + C$

Using limits gives the area between the curve and the $x$-axis.

## Question 1

Integrate the following expressions with respect to $x$.

**a)** $\\displaystyle\\int 2\\sqrt{x} - \\frac{1}{x^2} \\, dx$

**b)** $\\displaystyle\\int 4\\sqrt{x} - 2\\sqrt{x^5} \\, dx$

**c)** $\\displaystyle\\int \\frac{3}{4\\sqrt{x}} + \\frac{1}{x^3} \\, dx$

**d)** $\\displaystyle\\int 2x\\sqrt{x} - \\frac{4}{3x^2} \\, dx$

**e)** $\\displaystyle\\int 2\\sqrt{x} + \\frac{1}{2\\sqrt{x}} \\, dx$

Show Answers 

**a)** $\\dfrac{4}{3}x^{\\frac{3}{2}} + \\dfrac{1}{x} + C$

**b)** $\\dfrac{8}{3}x^{\\frac{3}{2}} - \\dfrac{4}{7}x^{\\frac{7}{2}} + C$

**c)** $\\dfrac{3}{2}\\sqrt{x} - \\dfrac{1}{2x^2} + C$

**d)** $\\dfrac{4}{5}x^{\\frac{5}{2}} + \\dfrac{4}{3}x^{-1} + C$

**e)** $\\dfrac{4}{3}x^{\\frac{3}{2}} + \\sqrt{x} + C$

## Question 2

The point $P(8,18)$ lies on the curve $C$, whose gradient function is given by $\\displaystyle\\frac{dy}{dx} = 8\\sqrt\[3\]{x} - 10, \\; x \\geq 0$.

Find an equation for $C$.

Show Answers 

$y = 6x^{\\frac{4}{3}} - 10x + 2$

### Question 3

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.23.29@2x.png)

Show Answers 

**a)** $(-5, 0)$ and $(3, 0)$

**b)** $-27$

### Question 4

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.36.52@2x.png)

Show Answers 

**a)** $(0,0)$, $(2,0)$ and $(3,0)$

**b)** Total area $= \\dfrac{8}{3} + \\dfrac{5}{12} = 3\\dfrac{1}{12}$

### Question 5

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.24.20@2x.png)

Show Answers 

**a)** $(1, 2)$ and $(3, 4)$

**b)** Area $= \\dfrac{4}{3}$

### Question 6

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.37.33@2x-1.png)

Show Answers 

**a)** $3$

**b)** Area $= 9$

### Question 7

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.26.27@2x.png)

Show Answers 

**a)** $y = 2 - x$

**b)** Area $= \\dfrac{5}{6}$

### Question 8

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/07/CleanShot-2026-07-23-at-16.25.43@2x.png)

Show Answers 

**a)** $y = 4 - x$

**b)** Area $= \\dfrac{5}{3}$

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