Integration

Intuition

Integration is the reverse of differentiation — it accumulates tiny pieces into a whole: If differentiation breaks things down (finds the rate of change), integration builds things up (finds the total). The area under a speed-time graph is distance travelled. The area under a power-time graph is energy used. Integration sums up infinitely many infinitesimal contributions to find a total.

Why it matters: Integration is used to find areas, volumes, work done, probability, and many other quantities that involve accumulation. It’s the second fundamental operation of calculus and the complement to differentiation.

The key insight: The Fundamental Theorem of Calculus connects differentiation and integration — they are inverse operations. If $F'(x) = f(x)$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. This means you can evaluate definite integrals by finding antiderivatives, without doing Riemann sums. This is the most important theorem in calculus.

Board Coverage

Board	Paper	Notes
AQA	Paper 1, 2	Basic integration in P1; by parts, substitution in P2
Edexcel	P1, P2	Similar split
OCR (A)	Paper 1, 2	Includes trapezium rule
CIE (9709)	P1, P2, P3	Basic in P1; by parts/substitution in P2/P3; further in P3

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1. Integration as Area: Riemann Sums

1.1 Definition

Definition. The definite integral of $f$ from $a$ to $b$ is defined as the limit of Riemann Sums:

$\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\,\Delta x$

Where $[a,b]$ is divided into $n$ subintervals of width $\Delta x = \dfrac{b-a}{n}$And $x_i^*$ is A sample point in the $i$-th subinterval.

Geometric picture. We divide the area under $y = f(x)$ between $x = a$ and $x = b$ into $n$ thin Rectangles. The sum of their areas approximates the total area. As $n \to \infty$ (rectangles become Infinitely thin), the approximation becomes exact.

• If we take the upper rectangle height (right endpoint), we get an upper sum.
• If we take the lower rectangle height (left endpoint), we get a lower sum.
• The integral exists when the upper and lower sums converge to the same limit.

Intuition. Integration is “accumulation” — adding up infinitely many infinitesimal Contributions. If $f(x)$ is a rate (like velocity), then $\int_a^b f(x)\,dx$ is the total change (displacement).

Increase the number of rectangles to see how Riemann sums converge to The exact area under the curve. Compare the upper and lower sums as the partition gets finer.

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2. The Fundamental Theorem of Calculus

Theorem (FTC). If $f$ is continuous on $[a,b]$Then

$(\mathrm{Part 1})\quad \frac{d}{dx}\int_a^x f(t)\,dt = f(x)$

$(\mathrm{Part 2})\quad \int_a^b f(x)\,dx = F(b) - F(a)$

Where $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$).

2.1 Sketch proof of Part 2

Let $G(x) = \int_a^x f(t)\,dt$. By Part 1, $G'(x) = f(x)$.

If $F$ is any other antiderivative of $f$Then $F'(x) = G'(x) = f(x)$So $F(x) - G(x) = C$ (a Constant).

$F(x) = G(x) + C \implies F(b) - F(a) = G(b) - G(a) = \int_a^b f(t)\,dt - 0$

Hence $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$. $\blacksquare$

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3. Standard Integrals

Each standard integral can be derived by reversing the corresponding differentiation.

3.1 Derivation of key standard integrals

Power rule. Since $\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = x^n$ for $n \neq -1$:

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$

Reciprocal. Since $\dfrac{d}{dx}\ln|x| = \dfrac{1}{x}$:

$\int \frac{1}{x}\,dx = \ln|x| + C$

Exponential. Since $\dfrac{d}{dx}e^{kx} = ke^{kx}$:

$\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$

Trigonometric. Since $\dfrac{d}{dx}\sin x = \cos x$:

$\int \cos x\,dx = \sin x + C$

$\int \sin x\,dx = -\cos x + C$

$\int \sec^2 x\,dx = \tan x + C$

3.2 Summary table

| $f(x)$ | $\int f(x)\,dx$ | | ------------------- | -------------------------- | --- | ---- | | $x^n$ ($n \neq -1$) | $\dfrac{x^{n+1}}{n+1} + C$ | | $1/x$ | $\ln | x | + C$ | | $e^{kx}$ | $\dfrac{1}{k}e^{kx} + C$ | | $\cos x$ | $\sin x + C$ | | $\sin x$ | $-\cos x + C$ | | $\sec^2 x$ | $\tan x + C$ |

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4. Definite Integration and Areas

4.1 Area under a curve

The area between $y = f(x)$The $x$-axis, $x = a$And $x = b$ is

$A = \int_a^b f(x)\,dx$

Provided $f(x) \geq 0$ on $[a,b]$.

4.2 Area between a curve and the $x$-axis

If $f(x)$ changes sign on $[a,b]$We must split the integral at each root:

$A = \int_a^b |f(x)|\,dx$

4.3 Area between two curves

The area between $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ (where $f(x) \geq g(x)$) is

$A = \int_a^b [f(x) - g(x)]\,dx$

4.4 Area under a parametric curve

$A = \int_{t_1}^{t_2} y\,\frac{dx}{dt}\,dt$

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5. Integration by Substitution

5.1 The method

Theorem. $\displaystyle\int f(g(x))g'(x)\,dx = F(g(x)) + C$ where $F' = f$.

5.2 Proof via the chain rule

Let $u = g(x)$. By the chain rule, $\dfrac{d}{dx}F(u) = F'(u)\dfrac{du}{dx} = f(u)\dfrac{du}{dx} = f(g(x))g'(x)$.

Therefore $\int f(g(x))g'(x)\,dx = F(g(x)) + C$. $\blacksquare$

In practice:

1. Choose a substitution $u = g(x)$.
2. Compute $du = g'(x)\,dx$.
3. Rewrite the integral entirely in terms of $u$.
4. Integrate, then substitute back.

Example. Find $\displaystyle\int 2x\sqrt{x^2+1}\,dx$.

Let $u = x^2 + 1$, $du = 2x\,dx$.

$\int 2x\sqrt{x^2+1}\,dx = \int \sqrt{u}\,du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x^2+1)^{3/2} + C$

Tip: Tip $\sqrt{g(x)}$ and $g'(x)$ in the integrand, try $u = g(x)$.

5.3 Definite integrals with substitution

For a definite integral, you can either:

• Substitute back to $x$ before evaluating, or
• Change the limits: when x = a$$u = g(a); when x = b$$u = g(b).

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6. Integration by Parts

6.1 The formula

Theorem. $\displaystyle\int u\,dv = uv - \int v\,du$

6.2 Proof via the product rule

From the product rule: $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$.

Integrating both sides:

$uv = \int u\frac{dv}{dx}\,dx + \int v\frac{du}{dx}\,dx$

$\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx$

I.e., $\displaystyle\int u\,dv = uv - \int v\,du$. $\blacksquare$

6.3 Choosing $u$ and $dv$

Use the mnemonic LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential). Choose $u$ From the leftmost category that appears.

Caution: Warning After applying the formula, swap $u$ and $dv$. Example. Find $\displaystyle\int x e^x\,dx$.

Let u = x$$dv = e^x\,dx. Then du = dx$$v = e^x.

$\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C = e^x(x-1) + C$

6.4 Reduction formulas

Integration by parts can produce reduction formulas relating $I_n$ to $I_{n-1}$.

Example. Find a reduction formula for $I_n = \displaystyle\int x^n e^x\,dx$.

Let u = x^n$$dv = e^x\,dx. Then du = nx^{n-1}\,dx$$v = e^x.

$I_n = x^n e^x - n\int x^{n-1} e^x\,dx = x^n e^x - nI_{n-1}$

This gives $I_n = x^n e^x - nI_{n-1}$Allowing us to reduce any $I_n$ to $I_0 = e^x + C$.

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7. The Trapezium Rule

7.1 Formula

To approximate $\displaystyle\int_a^b f(x)\,dx$Divide $[a,b]$ into $n$ equal strips of width $h = \dfrac{b-a}{n}$:

$\int_a^b f(x)\,dx \approx \frac{h}{2}\left[y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\right]$

Where $y_i = f(a + ih)$.

7.2 Derivation

Each strip is approximated by a trapezium. The area of the $i$-th trapezium is:

$A_i = \frac{h}{2}(y_{i-1} + y_i)$

Summing all trapezia:

\begin\{aligned\} \mathrm\{Total\} &= \frac\{h\}\{2\}(y_0+y_1) + \frac\{h\}\{2\}(y_1+y_2) + \cdots + \frac\{h\}\{2\}(y_\{n-1\}+y_n) \\ &= \frac\{h\}\{2\}\left[y_0 + 2y_1 + 2y_2 + \cdots + 2y_\{n-1\} + y_n\right] \end\{aligned\}

7.3 Error bound

If $f''$ is continuous on $[a,b]$ and $|f''(x)| \leq M$ for all $x \in [a,b]$Then the error $E$ Satisfies

$|E| \leq \frac{(b-a)^3}{12n^2}M$

Proof (sketch). For a single strip of width $h$The trapezium rule gives area $\dfrac{h}{2}[f(a) + f(a+h)]$While the true area is $\int_a^{a+h}f(x)\,dx$. By Taylor’s theorem, The error per strip is $-\dfrac{h^3}{12}f''(\xi)$ for some $\xi \in (a, a+h)$. Summing $n$ strips And using the bound $|f''| \leq M$:

$|E| \leq n \cdot \frac{h^3}{12} M = n \cdot \frac{(b-a)^3}{12n^3}M = \frac{(b-a)^3}{12n^2}M \quad \blacksquare$

Intuition. The error decreases as $1/n^2$ — doubling the number of strips quarters the error. The error also depends on how curved the function is (via $f''$). For a straight line ($f'' = 0$), The trapezium rule is exact.

Tip: Tip Concave-down functions and overestimates for concave-up functions.

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8. Further Techniques

8.1 Integrating $\dfrac{f'(x)}{f(x)}$

$\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$

This follows from the substitution $u = f(x)$.

8.2 Partial fractions

Rational functions can be integrated by first decomposing into partial fractions.

Example. $\displaystyle\int \frac{1}{x^2-1}\,dx = \int \frac{1}{2}\left(\frac{1}{x-1} - \frac{1}{x+1}\right)dx = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C$.

8.3 $\int \tan x\,dx$ and related

$\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx = -\ln|\cos x| + C = \ln|\sec x| + C$

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Problem Set

Problem 1

Evaluate $\displaystyle\int_0^2 (3x^2 - 4x + 1)\,dx$.

Solution 1

$\int_0^2 (3x^2 - 4x + 1)\,dx = \left[x^3 - 2x^2 + x\right]_0^2 = (8 - 8 + 2) - 0 = 2$

If you get this wrong, revise: Standard Integrals — Section 3.

Problem 2

Find the area enclosed between $y = x^2$ and $y = 2x$.

Solution 2

Intersection: $x^2 = 2x \implies x(x-2) = 0 \implies x = 0, 2$.

On $[0,2]$: $2x \geq x^2$ (since $2x - x^2 = x(2-x) \geq 0$).

$A = \int_0^2 (2x - x^2)\,dx = \left[x^2 - \frac{x^3}{3}\right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}$

If you get this wrong, revise: Definite Integration and Areas — Section 4.3.

Problem 3

Find $\displaystyle\int \frac{2x}{x^2+3}\,dx$.

Solution 3

Let $u = x^2 + 3$, $du = 2x\,dx$.

$\int \frac{2x}{x^2+3}\,dx = \int \frac{1}{u}\,du = \ln|u| + C = \ln(x^2+3) + C$

(Since $x^2+3 \gt 0$No absolute value needed.)

If you get this wrong, revise: Integration by Substitution — Section 5.

Problem 4

Evaluate $\displaystyle\int_0^{\pi/2} x\sin x\,dx$.

Solution 4

Integration by parts: u = x$$dv = \sin x\,dx. Then du = dx$$v = -\cos x.

$\int x\sin x\,dx = -x\cos x + \int \cos x\,dx = -x\cos x + \sin x + C$

$\left[-x\cos x + \sin x\right]_0^{\pi/2} = \left(0 + 1\right) - \left(0 + 0\right) = 1$

If you get this wrong, revise: Integration by Parts — Section 6.

Problem 5

Use the trapezium rule with 4 strips to approximate $\displaystyle\int_0^2 \frac{1}{1+x^2}\,dx$And give an error bound given $|f''(x)| \leq 2$ on $[0,2]$.

Solution 5

$h = \dfrac{2-0}{4} = 0.5$. Values: y_0 = 1$$y_1 = 1/(1+0.25) = 0.8$$y_2 = 1/(1+1) = 0.5$$y_3 = 1/(1+2.25) \approx 0.3077$$y_4 = 1/5 = 0.2.

$\mathrm{Approx} = \frac{0.5}{2}\left[1 + 2(0.8) + 2(0.5) + 2(0.3077) + 0.2\right] = 0.25[1 + 1.6 + 1.0 + 0.6154 + 0.2] = 0.25 \times 4.4154 \approx 1.104$

Error bound: $|E| \leq \dfrac{2^3}{12 \times 16} \times 2 = \dfrac{8}{192} \times 2 = \dfrac{1}{12} \approx 0.0833$.

If you get this wrong, revise: The Trapezium Rule — Section 7.

Problem 6

Find $\displaystyle\int \frac{x}{\sqrt{x+1}}\,dx$.

Solution 6

Let $u = x+1$So $x = u-1$ and $dx = du$.

$\int \frac{u-1}{\sqrt{u}}\,du = \int(u^{1/2} - u^{-1/2})\,du = \frac{2}{3}u^{3/2} - 2u^{1/2} + C = \frac{2}{3}(x+1)^{3/2} - 2(x+1)^{1/2} + C$

If you get this wrong, revise: Integration by Substitution — Section 5.

Problem 7

Find a reduction formula for $I_n = \displaystyle\int_0^{\pi/2} \sin^n x\,dx$ for $n \geq 2$.

Solution 7

$I_n = \int_0^{\pi/2}\sin^{n-1}x \cdot \sin x\,dx$.

Let u = \sin^{n-1}x$$dv = \sin x\,dx. du = (n-1)\sin^{n-2}x\cos x\,dx$$v = -\cos x.

$I_n = \left[-\sin^{n-1}x\cos x\right]_0^{\pi/2} + (n-1)\int_0^{\pi/2}\sin^{n-2}x\cos^2 x\,dx$

The boundary term vanishes (since $\cos(\pi/2) = 0$ and $\sin 0 = 0$). Using $\cos^2 x = 1 - \sin^2 x$:

$I_n = (n-1)\int_0^{\pi/2}\sin^{n-2}x(1-\sin^2 x)\,dx = (n-1)(I_{n-2} - I_n)$

$I_n = (n-1)I_{n-2} - (n-1)I_n$ $nI_n = (n-1)I_{n-2}$ $I_n = \frac{n-1}{n}I_{n-2}$

If you get this wrong, revise: Reduction Formulas — Section 6.4.

Problem 8

Find the total area between $y = x(x-2)(x+1)$ and the $x$-axis.

Solution 8

$y = x(x-2)(x+1) = x^3 - x^2 - 2x$. Roots at $x = -1, 0, 2$.

$\int_{-1}^0 (x^3-x^2-2x)\,dx = \left[\frac{x^4}{4}-\frac{x^3}{3}-x^2\right]_{-1}^0 = 0 - \left(\frac{1}{4}+\frac{1}{3}-1\right) = -\left(\frac{3+4-12}{12}\right) = \frac{5}{12}$

$\int_0^2 (x^3-x^2-2x)\,dx = \left[\frac{x^4}{4}-\frac{x^3}{3}-x^2\right]_0^2 = \left(4-\frac{8}{3}-4\right) - 0 = -\frac{8}{3}$

Total area = $\dfrac{5}{12} + \dfrac{8}{3} = \dfrac{5+32}{12} = \dfrac{37}{12}$.

If you get this wrong, revise: Definite Integration and Areas — Section 4.2.

Problem 9

Find $\displaystyle\int e^{2x}\cos x\,dx$.

Solution 9

Apply integration by parts twice. Let $I = \int e^{2x}\cos x\,dx$.

First: u = e^{2x}$$dv = \cos x\,dx. du = 2e^{2x}\,dx$$v = \sin x.

$I = e^{2x}\sin x - 2\int e^{2x}\sin x\,dx$.

Second: u = e^{2x}$$dv = \sin x\,dx. du = 2e^{2x}\,dx$$v = -\cos x.

$\int e^{2x}\sin x\,dx = -e^{2x}\cos x + 2\int e^{2x}\cos x\,dx = -e^{2x}\cos x + 2I$.

$I = e^{2x}\sin x - 2(-e^{2x}\cos x + 2I) = e^{2x}\sin x + 2e^{2x}\cos x - 4I$.

$5I = e^{2x}(\sin x + 2\cos x)$.

$I = \frac{e^{2x}(\sin x + 2\cos x)}{5} + C$

If you get this wrong, revise: Integration by Parts — Section 6.

Problem 10

Evaluate $\displaystyle\int_1^e \frac{\ln x}{x}\,dx$.

Solution 10

Let $u = \ln x$, $du = \dfrac{1}{x}\,dx$.

$\int \frac{\ln x}{x}\,dx = \int u\,du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C$

$\left[\frac{(\ln x)^2}{2}\right]_1^e = \frac{1}{2} - 0 = \frac{1}{2}$

If you get this wrong, revise: Integration by Substitution — Section 5.

Problem 11

The curve $C$ has parametric equations $x = t^2$, $y = 2t$ for $0 \leq t \leq 3$. Find the area under $C$.

Solution 11

$A = \int_{t=0}^{t=3} y\,\frac{dx}{dt}\,dt = \int_0^3 2t \cdot 2t\,dt = \int_0^3 4t^2\,dt = \left[\frac{4t^3}{3}\right]_0^3 = \frac{4 \times 27}{3} = 36$

If you get this wrong, revise: Definite Integration and Areas — Section 4.4.

Problem 12

Find $\displaystyle\int \frac{3x+5}{x^2+4x+3}\,dx$.

Solution 12

$x^2 + 4x + 3 = (x+1)(x+3)$.

$\frac{3x+5}{(x+1)(x+3)} = \frac{A}{x+1} + \frac{B}{x+3}$

$3x + 5 = A(x+3) + B(x+1)$.

$x = -1$: $2 = 2A \implies A = 1$. $x = -3$: $-4 = -2B \implies B = 2$.

$\int \frac{1}{x+1} + \frac{2}{x+3}\,dx = \ln|x+1| + 2\ln|x+3| + C$

If you get this wrong, revise: Partial Fractions — Section 8.2.

Problem 13

Sketch proof: explain why $\displaystyle\int_{-a}^a f(x)\,dx = 0$ when $f$ is an odd function.

Solution 13

If $f$ is odd, then $f(-x) = -f(x)$.

$\int_{-a}^a f(x)\,dx = \int_{-a}^0 f(x)\,dx + \int_0^a f(x)\,dx$

Let $u = -x$ in the first integral:

$\int_{-a}^0 f(x)\,dx = \int_a^0 f(-u)(-du) = \int_0^a -f(u)\,du = -\int_0^a f(u)\,du$

Therefore $\displaystyle\int_{-a}^a f(x)\,dx = -\int_0^a f(x)\,dx + \int_0^a f(x)\,dx = 0$. $\blacksquare$

If you get this wrong, revise: Definite Integration and Areas — Section 4.

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Common Pitfalls

1. Forgetting the chain rule when integrating composite functions. Look for an inner function and its derivative.

2. Confusing definite and indefinite integrals. Definite integrals give a numerical value; indefinite integrals give a family of functions.

3. Incorrectly applying limits of integration after a substitution. Remember to change the limits to the new variable.

4. Losing marks by not showing sufficient working. Always write out each step, especially in proof questions.

5. Misreading the question, particularly with ‘hence’ vs ‘hence or otherwise’. The former requires using previous work.

6. Forgetting to check that solutions satisfy the original equation (especially with squaring both sides or dividing by variables).

Summary

The key principles covered in this topic are linked in the sub-pages above. Focus on understanding the definitions, applying the formulas or frameworks, and evaluating strengths and limitations of each approach.

Worked Examples

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

Cross-References

• Differentiation — Integration reverses differentiation, and the Fundamental Theorem of Calculus connects rates of change to accumulated areas.
• Trigonometry — Trigonometric identities are essential for integrating functions involving $\sin$, $\cos$, and $\tan$.
• Numerical Methods — The trapezium rule and Simpson’s rule approximate definite integrals when exact integration is difficult.
• Exponentials and Logarithms — The integral of $1/x$ gives $\ln|x|$, linking integration directly to logarithmic functions.