New 2026 Level 3 Integration prediction exam is out: a full paper with worked answers, free. Try it

L3 Integration

Question Bank

Basic Integrals Reverse Chain, Logs & Division Trig Identities & Products Definite Integrals & Unknowns Areas Kinematics & Rates of Change dy/dx = f(x): Finding y Separable Differential Equations Growth, Decay & Cooling Trapezium & Simpson’s Rules Unfamiliar Contexts
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Find $\displaystyle\int \left(4x^3 - \frac{6}{x^4}\right) dx$.

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Find $\displaystyle\int \left(e^x + 8e^{-2x}\right) dx$.

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Find $\displaystyle\int 6\sec^2 3x\,dx$.

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Find $\displaystyle\int \left(6x^2 + \frac{5}{x} + 4e^{2x}\right) dx$.

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Find $\displaystyle\int f(x)\,dx$, where $f(x) = \dfrac{12}{3x + 4}$.

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Find $\displaystyle\int (6\cos 3x - 4\sin 2x)\,dx$.

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Find $\displaystyle\int \left(9t^{\frac{1}{2}} + 4t^{-\frac{5}{3}}\right) dt$.

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Find $\displaystyle\int \left(e^2 + 2e^{\frac{x}{2}}\right) dx$.

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Find $\displaystyle\int \sec\left(\frac{x}{2}\right)\tan\left(\frac{x}{2}\right) dx$.

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Find $\displaystyle\int \left(0.6t^2 + 4e^{0.2t}\right) dt$.

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Find $\displaystyle\int \left(8\sin 4x + \frac{3}{x^2} - 1\right) dx$.

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Find $\displaystyle\int \left(\frac{2}{5x} - \frac{5}{x^2}\right) dx$.

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Find $\displaystyle\int \left((2x + 1)^4 + \frac{6}{2x + 1}\right) dx$.

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Find $\displaystyle\int \frac{8}{(4x - 3)^3}\,dx$.

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Find $\displaystyle\int 8\operatorname{cosec} 4x\cot 4x\,dx$.

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Find $\displaystyle\int \frac{6}{\sqrt{3x + 1}}\,dx$.

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Find $\displaystyle\int 12\sin(1 - 4x)\,dx$.

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Find $\displaystyle\int (2x^3 - 1)^2\,dx$.

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Find $\displaystyle\int \left(6\sqrt[3]{x^2} - \frac{4}{\sqrt[3]{x}}\right) dx$.

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Find $\displaystyle\int \left(5\cos\left(\frac{x}{5}\right) - 2e^{-\frac{x}{2}}\right) dx$.

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Find $\displaystyle\int 4\operatorname{cosec}^2\left(\frac{t}{2}\right) dt$.

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Find $\displaystyle\int \sec x(\sec x + \tan x)\,dx$.

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Find $\displaystyle\int \frac{x^3 + 4x - 6}{x^2}\,dx$.

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Find $\displaystyle\int \left(\sqrt{e^x} + \frac{1}{e^{2x}}\right) dx$.

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Find $\displaystyle\int \left(\frac{3}{x} - \frac{8}{1 - 4x}\right) dx$.

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Find $\displaystyle\int \left(6e^{3x - 2} - 4\sin(\pi x)\right) dx$.

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Find $\displaystyle\int \frac{4x - 1}{2\sqrt{x}}\,dx$.

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

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Find $\displaystyle\int f(x)\,dx$, where

$ \begin{aligned}f(x) &= 4\sec^2 2x\\[0pt]&\quad {}+ 9\operatorname{cosec}(1 - 3x)\\[0pt]&\qquad {}\times \cot(1 - 3x).\end{aligned} $

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Find $\displaystyle\int 3\sec(2 - x)\tan(2 - x)\,dx$.

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Find $\displaystyle\int \left(1 + \frac{1}{\sqrt{x}}\right)^2 dx$.