Sec 4 · Additional Mathematics

Integration

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1. Integrating powers of x

Rule	Formula	Example
power ( $n \neq -1$ )	$\int x^{n} \, dx = \dfrac{x^{n + 1}}{n + 1} + c$	$\int x^{4} \, dx = \dfrac{x^{5}}{5} + c$
constant	$\int k \, dx = kx + c$	$\int 7 \, dx = 7x + c$
sum	integrate term by term	$\int (3x^{2} - 4x) \, dx = x^{3} - 2x^{2} + c$

Write as a power of x first

\displaystyle \int \dfrac{4}{x^{2}} \, dx

$\int 4x^{-2} \, dx = \dfrac{4x^{-1}}{-1} + c$
$\int \dfrac{4}{x^{2}} \, dx = -\dfrac{4}{x} + c$

📖 Worked example

Find $\int (x - 4)(3x + 1) \, dx$ .

Step 1 of 2

Expand the brackets first: integrate term by term

\int (x - 4)(3x + 1) \, dx = \int \left(3x^{2} - 11x - 4\right) \, dx

Practise this →

2. Integrating (ax + b)ⁿ

Rule	Formula	Example
power of $ax + b$	$\int (ax + b)^{n} \, dx = \dfrac{(ax + b)^{n + 1}}{a(n + 1)} + c$	$\int (2x + 1)^{4} \, dx = \dfrac{(2x + 1)^{5}}{10} + c$

📖 Worked example

Find $\int \dfrac{4}{\sqrt{4x + 3}} \, dx$ .

Step 1 of 3

Write it as a power of the bracket

\int 4(4x + 3)^{-\frac{1}{2}} \, dx

Practise this →

3. Trigonometric, exponential and 1 over ax + b

$f(x)$	$\int f(x) \, dx$	Example
$\cos(ax + b)$	$\dfrac{1}{a}\sin(ax + b) + c$	$\int \cos 2x \, dx = \dfrac{1}{2}\sin 2x + c$
$\sin(ax + b)$	$-\dfrac{1}{a}\cos(ax + b) + c$	$\int \sin 3x \, dx = -\dfrac{1}{3}\cos 3x + c$
$\sec^{2}(ax + b)$	$\dfrac{1}{a}\tan(ax + b) + c$	$\int \sec^{2} 2x \, dx = \dfrac{1}{2}\tan 2x + c$
$e^{ax + b}$	$\dfrac{1}{a}e^{ax + b} + c$	$\int e^{3x + 1} \, dx = \dfrac{1}{3}e^{3x + 1} + c$
$\dfrac{1}{ax + b}$	$\dfrac{1}{a}\ln(ax + b) + c$ , $ax + b > 0$	$\int \dfrac{4}{2x + 1} \, dx = 2\ln(2x + 1) + c$

📖 Worked example

Find $\int \dfrac{4}{2x + 1} \, dx$ , where $x > -\dfrac{1}{2}$ .

Step 1 of 2

Recall the integral of 1/x (check by differentiating it back)

\text{The integral of 1/x is ln x.}

Practise this →

4. Finding the equation of a curve

Gradient 3x² − 4x, through (2, 5)

\displaystyle y = x^{3} - 2x^{2} + c

e.g.

\displaystyle 5 = 2^{3} - 2(2)^{2} + c

\displaystyle c = 5

$y = x^{3} - 2x^{2} + 5$

📖 Worked example

The gradient of a curve is given by $\dfrac{dy}{dx} = 2e^{2x} + 3$ . The curve passes through the point $(0, 1)$ . Find the equation of the curve.

Step 1 of 3

Integrate: eᵇˣ integrates to eᵇˣ ÷ b, and a number k to kx. Add c

y = e^{2x} + 3x + c

Practise this →

5. Integration as the reverse of differentiation

Given	Then
$\dfrac{d}{dx}F(x) = f(x)$	$\int f(x) \, dx = F(x) + c$
$\dfrac{d}{dx}(x^{2} + 1)^{4} = 8x(x^{2} + 1)^{3}$	$\int x(x^{2} + 1)^{3} \, dx = \dfrac{1}{8}(x^{2} + 1)^{4} + c$

📖 Worked example

Given that $\dfrac{d}{dx}\left(x^{2}\ln x\right) = 2x\ln x + x$ , find $\int x\ln x \, dx$ .

Step 1 of 4

Integrate both sides of the given result: integration reverses differentiation

x^{2}\ln x = \int 2x\ln x \, dx + \int x \, dx

Practise this →

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Sec 4 · Additional Mathematics

Integration

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1. Integrating powers of x

Rule	Formula	Example
power ( $n \neq -1$ )	$\int x^{n} \, dx = \dfrac{x^{n + 1}}{n + 1} + c$	$\int x^{4} \, dx = \dfrac{x^{5}}{5} + c$
constant	$\int k \, dx = kx + c$	$\int 7 \, dx = 7x + c$
sum	integrate term by term	$\int (3x^{2} - 4x) \, dx = x^{3} - 2x^{2} + c$

Write as a power of x first

\displaystyle \int \dfrac{4}{x^{2}} \, dx

$\int 4x^{-2} \, dx = \dfrac{4x^{-1}}{-1} + c$
$\int \dfrac{4}{x^{2}} \, dx = -\dfrac{4}{x} + c$

📖 Worked example

Find $\int (x - 4)(3x + 1) \, dx$ .

Step 1 of 2

Expand the brackets first: integrate term by term

\int (x - 4)(3x + 1) \, dx = \int \left(3x^{2} - 11x - 4\right) \, dx

Practise this →

2. Integrating (ax + b)ⁿ

Rule	Formula	Example
power of $ax + b$	$\int (ax + b)^{n} \, dx = \dfrac{(ax + b)^{n + 1}}{a(n + 1)} + c$	$\int (2x + 1)^{4} \, dx = \dfrac{(2x + 1)^{5}}{10} + c$

📖 Worked example

Find $\int \dfrac{4}{\sqrt{4x + 3}} \, dx$ .

Step 1 of 3

Write it as a power of the bracket

\int 4(4x + 3)^{-\frac{1}{2}} \, dx

Practise this →

3. Trigonometric, exponential and 1 over ax + b

$f(x)$	$\int f(x) \, dx$	Example
$\cos(ax + b)$	$\dfrac{1}{a}\sin(ax + b) + c$	$\int \cos 2x \, dx = \dfrac{1}{2}\sin 2x + c$
$\sin(ax + b)$	$-\dfrac{1}{a}\cos(ax + b) + c$	$\int \sin 3x \, dx = -\dfrac{1}{3}\cos 3x + c$
$\sec^{2}(ax + b)$	$\dfrac{1}{a}\tan(ax + b) + c$	$\int \sec^{2} 2x \, dx = \dfrac{1}{2}\tan 2x + c$
$e^{ax + b}$	$\dfrac{1}{a}e^{ax + b} + c$	$\int e^{3x + 1} \, dx = \dfrac{1}{3}e^{3x + 1} + c$
$\dfrac{1}{ax + b}$	$\dfrac{1}{a}\ln(ax + b) + c$ , $ax + b > 0$	$\int \dfrac{4}{2x + 1} \, dx = 2\ln(2x + 1) + c$

📖 Worked example

Find $\int \dfrac{4}{2x + 1} \, dx$ , where $x > -\dfrac{1}{2}$ .

Step 1 of 2

Recall the integral of 1/x (check by differentiating it back)

\text{The integral of 1/x is ln x.}

Practise this →

4. Finding the equation of a curve

Gradient 3x² − 4x, through (2, 5)

\displaystyle y = x^{3} - 2x^{2} + c

e.g.

\displaystyle 5 = 2^{3} - 2(2)^{2} + c

\displaystyle c = 5

$y = x^{3} - 2x^{2} + 5$

📖 Worked example

The gradient of a curve is given by $\dfrac{dy}{dx} = 2e^{2x} + 3$ . The curve passes through the point $(0, 1)$ . Find the equation of the curve.

Step 1 of 3

Integrate: eᵇˣ integrates to eᵇˣ ÷ b, and a number k to kx. Add c

y = e^{2x} + 3x + c

Practise this →

5. Integration as the reverse of differentiation

Given	Then
$\dfrac{d}{dx}F(x) = f(x)$	$\int f(x) \, dx = F(x) + c$
$\dfrac{d}{dx}(x^{2} + 1)^{4} = 8x(x^{2} + 1)^{3}$	$\int x(x^{2} + 1)^{3} \, dx = \dfrac{1}{8}(x^{2} + 1)^{4} + c$

📖 Worked example

Given that $\dfrac{d}{dx}\left(x^{2}\ln x\right) = 2x\ln x + x$ , find $\int x\ln x \, dx$ .

Step 1 of 4

Integrate both sides of the given result: integration reverses differentiation

x^{2}\ln x = \int 2x\ln x \, dx + \int x \, dx

Practise this →

Try a question: Integration

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Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Practise integration with step-by-step help

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