Integration

Explanation

• Integration is the reverse process of differentiation: if $\frac{\mathrm dy}{\mathrm dx}=f(x)$, then $y=\int f(x)\,\mathrm dx$ recovers the family of curves with that gradient function.
• Every indefinite integral needs a constant of integration: $\int f(x)\,\mathrm dx=F(x)+c$.
• One known point on the curve fixes the value of $c$.
• A definite integral is evaluated from any antiderivative: $\int_a^b f(x)\,\mathrm dx=F(b)-F(a)$; the constant of integration cancels in the subtraction.
• A common error is to omit $+c$ from an indefinite integral, or to substitute the known point into $f(x)$ instead of the integrated function when finding $c$.

Explanation

• For $n\ne-1$, $\int xⁿ\,\mathrm dx=\frac{x^{n+1}}{n+1}+C$; rewrite roots and reciprocals as powers before applying the rule. Standard forms include $\int e^{kx}\,\mathrm dx=\frac1k e^{kx}+C$ and $\int\frac1x\,\mathrm dx=\ln|x|+C$ on intervals not crossing $0$.
• Divide by the inner coefficient in trigonometric integrals.
• Also use $\int\sec^2(kx)\,\mathrm dx=\frac1k\tan(kx)+C$, $\int\tan x\,\mathrm dx=-\ln|\cos x|+C$, and identities for powers such as $\sin^2x=\frac12(1-\cos2x)$.
• A common error is to forget the constant of integration or to multiply by $k$ rather than divide by it when integrating a function of $kx$.
• Integrate term by term: the first three terms give $2e^{3x}+4\ln|x|+\frac52\cos(2x)$.

Explanation

• $\int_a^b f(x)\,dx=F(b)-F(a)$ where $F$ is an antiderivative; write the antiderivative in square brackets with the limits, substitute both, and subtract in the right order, using $e^0=1$ and $\ln1=0$ where they appear.
• The integral is a signed area, so a region below the $x$-axis integrates to a negative number and its area is the modulus; split at the roots when a region crosses the axis.
• For the area between a curve and a line or a second curve, find the intersections first, then integrate $y_{\text{upper}}-y_{\text{lower}}$ between them, or subtract the area of a triangle or trapezium under a straight line from the area under the curve.
• Exact answers keep surds, $\pi$, $e$ and logarithms; “show all stages of working” means calculator-only answers score nothing.
• A parametric curve has area $\int y\,\dfrac{dx}{dt}\,dt$ with the limits in $t$.

Explanation

• Partition an interval into strips of width $\Delta x$ and form $\sum f(x_r)\Delta x$; its limit as the maximum strip width tends to zero is the definite integral.
• To recognise a limit, identify the factor playing the role of $\Delta x$ and rewrite the sampled expression in terms of an endpoint such as $x_r=a+r\Delta x$.
• A sum with factor $1/n$ usually samples an interval of length $1$; a different interval width introduces a corresponding scale factor.
• A common error is to identify the integrand but omit the width factor, which changes the value of the limiting integral.

Explanation

• Substitution reverses the chain rule: choose $u=g(x)$ so the remaining factor supplies $\mathrm du=g'(x)\,\mathrm dx$. In particular, recognise $\int\frac{f'(x)}{f(x)}\,\mathrm dx=\ln|f(x)|+C$.
• Integration by parts reverses the product rule: $\int u\,\mathrm dv=uv-\int v\,\mathrm du$; choose $u$ so that differentiating it simplifies the integral.
• For a definite integral, either change the limits to the new variable or return fully to the original variable before applying the old limits.
• A common error is to omit the minus sign in integration by parts or to mix $x$ and $u$ in the same transformed integral.
• A required standard result is $\int\ln x\,dx=x\ln x-x+c$, obtained by integration by parts with $u=\ln x$ and $dv=dx$.

Explanation

• Factor the denominator fully into linear factors, then write one partial-fraction term for each factor before solving for its constants. For $\frac{A}{x-a}$, the integral is $A\ln|x-a|+C$; retain absolute values unless the domain makes the sign known.
• Determine constants by equating coefficients or substituting convenient values that make all but one term vanish.
• A common error is to integrate the unfactorised denominator as though $\int f'(x)/f(x)\,\mathrm dx$ applied when the numerator is not its derivative.
• Set $\frac{5x+1}{(x-1)(x+2)}=\frac A{x-1}+\frac B{x+2}$.
• Then $5x+1=A(x+2)+B(x-1)$.

Explanation

• For $\frac{\mathrm dy}{\mathrm dx}=g(x)h(y)$, rearrange to place all $y$-terms with $\mathrm dy$ and all $x$-terms with $\mathrm dx$, then integrate both sides.
• Include a constant after integration and use the initial condition to determine it.
• A family of solutions can be sketched from its equilibrium curves, initial values, gradients and long-term behaviour.
• When logarithms arise, exponentiate carefully and use the stated domain or initial condition to choose any required sign or branch.
• A common error is to divide by a factor involving $y$ without checking whether doing so loses a constant equilibrium solution.

Explanation

• Interpret constants and initial values with their units, and examine limiting behaviour to identify equilibrium values or long-term predictions.
• In kinematics, $v=\frac{\mathrm ds}{\mathrm dt}$ and $a=\frac{\mathrm dv}{\mathrm dt}$; the sign of velocity gives direction, while a change of sign marks a reversal of motion.
• Check whether the mathematical solution remains meaningful on the time interval and within the physical range assumed by the model.
• A common error is to state only that a model is 'unrealistic'; name a specific assumption, such as constant environmental conditions or neglect of resistance, and explain its effect.