Integration

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1. 1Integration techniquesThe curve from its gradient, simplifying before integrating, substitution (and changing the limits), logarithms, and splitting into partial fractions first.Done20 min
2. 2Areas, volumes and motionAreas that lie below the x-axis, areas between curves, volumes of revolution about either axis, and going from acceleration to velocity to distance by integrating.Done21 min
3. 3The trapezium ruleEstimate a definite integral from a table of values: strips, ordinates and the rule ½h[first + last + 2(the rest)], and whether the estimate is too big or too small.Done14 min

Past questions (41)

From 2008–2023: 39 WAEC, 2 NECO. Theory questions first, then objective, newest first.

Theory (35)

• NECO 2023 · Paper 2 · Q4Using the trapezium rule with 7 ordinates $x = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5$ and $5.0$, estimate the value of $\displaystyle\int_2^5 (2x + 3)^2\,dx$.
• WAEC 2023 · Paper 2 · Q2Using the trapezium rule with five ordinates, evaluate, correct to two decimal places, $\displaystyle\int_0^2 \frac{4}{1 + x^2}\,dx$.
• WAEC 2023 · Paper 2 · Q8The acceleration, $a$, of a particle starting from rest and moving at any time $t$ seconds is given by $a = (20t - 3t^2)\text{ m s}^{-2}$. Find …
• WAEC 2022 · Paper 2 · Q14A particle initially at rest moves in a straight line with an acceleration of $(10t - 4t^2)\text{ m s}^{-2}$. Find the: (i) velocity of the particle …
• WAEC 2022 · Paper 2 · Q3Using the trapezium rule with five ordinates, evaluate, correct to two decimal places, $\displaystyle\int_1^3 \frac{2}{x + 4}\,dx$.
• WAEC 2020 · Paper 2 · Q11Find the positive value of $y$ for which $\begin{vmatrix} y & 1 & 0 \\ 1 & y & 1 \\ 0 & 1 & y \end{vmatrix} = 0$ (4 d.p.).
• WAEC 2019 · Paper 2 · Q10If $\sin p = \frac12$ and $\cos q = -\frac13$, evaluate $\sin(p - q)$, where $0^\circ \le p \le 90^\circ$ and $90^\circ \le q \le 180^\circ$.
• WAEC 2019 · Paper 2 · Q9If $y = 2x^2 - 5x + 3$ is expressed in the form $y = p(x + q)^2 + r$, where $p$, $q$ and $r$ are constants, find $(q + r)$.
• WAEC 2018 · Paper 2 · Q10Solve $2^{3y + 2} - 7(2^{2y + 2}) - 31(2^y) - 8 = 0$, $y \in \mathbb R$.
• WAEC 2018 · Paper 2 · Q2Using the trapezium rule with ordinates at $x = 1, 2, 3, 4$ and $5$, calculate, correct to two decimal places, the value of $\displaystyle\int_1^5 \left(x + \frac{2}{x^2}\right)dx$.
• WAEC 2018 · Paper 2 · Q11Find, from first principles, the derivative of $5x - \dfrac{6}{x^2}$ with respect to $x$.
• WAEC 2018 · Paper 2 · Q2If $\displaystyle\int_1^t (2x + 5)\,dx = 18$, find the value of $t$.
• WAEC 2017 · Paper 2 · Q11If $f(x) = \displaystyle\int(4x - x^2)\,dx$ and $f(3) = 21$, find $f(x)$.
• WAEC 2017 · Paper 2 · Q2Evaluate $\displaystyle\int_1^3 \frac{x - 1}{(x + 1)^2}\,dx$ (3 d.p.).
• WAEC 2017 · Paper 2 · Q11Find the values of $x$ at the points of intersection of the curves $y = (2x - 5)^3$ and $y = \dfrac{1}{(2x - 5)^3}$.
• WAEC 2017 · Paper 2 · Q4Given that $y = 3x^2 - \dfrac{3}{x^2}$, $x \ne 0$, find:
• WAEC 2016 · Paper 2 · Q10(i) Write down the expansion of $(1 + x)^7$ in ascending powers of $x$. (ii) If the coefficients of the fifth, sixth and seventh terms in the …
• WAEC 2016 · Paper 2 · Q9Express $\dfrac{x + 6}{(x + 1)^3}$ in partial fractions.
• WAEC 2014 · Paper 2 · Q11If $f(x) = \dfrac{x - 3}{2x - 1}$, $x \ne \frac12$ and $g(x) = \dfrac{x - 1}{x + 1}$, $x \ne -1$, find $g \circ f$.
• WAEC 2014 · Paper 2 · Q10If $P = \begin{pmatrix} 4 & 2 \\ c & 3 \end{pmatrix}$, $Q = \begin{pmatrix} 6 & 2 \\ 4 & d \end{pmatrix}$ and $PQ = QP$, find the values of $c$ …
• WAEC 2014 · Paper 2 · Q9Resolve $\dfrac{x^2 + 1}{(x + 2)^3}$ into partial fractions.
• WAEC 2013 · Paper 2 · Q11The sum of the first three terms of a decreasing exponential sequence (G.P.) is equal to 7 and the product of these three terms is equal to 8. …
• WAEC 2011 · Paper 2 · Q9Find, from first principles, the derivative of $f(x) = (2x + 3)^2$.
• WAEC 2011 · Paper 2 · Q10The gradient of a curve is given by $2x - 3x^2$. Find the equation of the curve if the point $(1, 2)$ lies on it.
• WAEC 2011 · Paper 2 · Q12Given the curve $y = x^2 - 4$, calculate, correct to two decimal places, the:
• WAEC 2011 · Paper 2 · Q9Use the trapezium rule with ordinates at $x = 1, 2, 3, 4$ and $5$ to calculate, correct to two decimal places, an approximate value for $\displaystyle\int_1^5 (2x + 8x^{-2})\,dx$.
• WAEC 2010 · Paper 2 · Q10If $x^2 + y^2 = py(1 + x^2)$, where $p$ is a constant, find $\dfrac{dy}{dx}$.
• WAEC 2010 · Paper 2 · Q11Find the area enclosed by the $x$-axis and the curve $y = 3x^2 + 2x - 1$.
• WAEC 2010 · Paper 2 · Q12Given that $\displaystyle\int_1^m (x^2 - 2x + 1)\,dx = \dfrac13$, $m > 0$, determine the value of $m$.
• WAEC 2009 · Paper 2 · Q10Using the same axes, sketch the curves $y = 6 - x - x^2$ and $y = 3x^2 - 2x + 3$.
• WAEC 2009 · Paper 2 · Q11Evaluate $\displaystyle\int_1^2 \frac{x}{\sqrt{5 - x^2}}\,dx$.
• WAEC 2009 · Paper 2 · Q12Use the trapezium rule with five ordinates to evaluate $\displaystyle\int_0^1 \frac{x}{1 + x^2}\,dx$, correct to four significant figures.
• WAEC 2009 · Paper 2 · Q12Two linear transformations are defined by $P : (x, y) \to (-2x + 3y,\ 5x - 4y)$ and $T : (x, y) \to (-3x - 4y,\ 6x - 5y)$. Find the inverse of …
• WAEC 2008 · Paper 2 · Q12Given that $M\begin{pmatrix} 2 & 9 \\ 8 & 6 \end{pmatrix} = \begin{pmatrix} 14 & 3 \end{pmatrix}$, find the matrix $M$.
• WAEC 2008 · Paper 2 · Q9A curve has gradient $3x^2 - 2x + 1$ at the point $(x, y)$. If it passes through the point $(1, 3)$, find its equation.

Objective (6)

• NECO 2023 · Paper 1 · Q17A curve passes through $(-2, 3)$ and its gradient at any point is $3x^2 + 8x - 5$. Find the equation of the curve.
• WAEC 2023 · Paper 1 · Q34Evaluate: $\displaystyle\int (2x + 1)^3\,dx$.
• WAEC 2023 · Paper 1 · Q5Evaluate: $\displaystyle\int_0^1 x(x^2 - 2)^2\,dx$.
• WAEC 2022 · Paper 1 · Q10Evaluate: $\displaystyle\int_0^1 x^2\left(x^3 + 2\right)^3 dx$.
• WAEC 2022 · Paper 1 · Q34The gradient of a function at any point $(x, y)$ is $2x - 6$. If the function passes through $(1, 2)$, find the function.
• WAEC 2020 · Paper 1 · Q4If $\displaystyle\int_0^3 (px^2 + 16)\,dx = 129$, find the value of $p$.