Browse Integration

313 questions at your level

DifficultyAllLevel 1Level 2Level 3Level 4Level 5

Indefinite integration (reversing differentiation)

22 questions

LessonNot started
Find

∫(8x3−3x2+5x)dx\int \left(8x^3 - \frac{3}{x^2} + 5\sqrt{x}\right)dx

writing each term in simplest form.

(4 marks)
●●●●●Level 24 marksStart
Find ∫3x2 dx\displaystyle\int 3x^2 \,\mathrm{d}x.
●●●●●Level 22 marksStart
Find

∫(x2−3)(2+5x2)dx\int \left(x^2 - 3\right)\left(2 + \frac{5}{x^2}\right)\mathrm{d}x

writing your answer in simplest form.

(4 marks)
●●●●●Level 24 marksStart

The constant of integration

17 questions

LessonNot started
Find

∫(8x3−3x2+5x)dx\int \left(8x^3 - \frac{3}{x^2} + 5\sqrt{x}\right)dx

writing each term in simplest form.

(4 marks)
●●●●●Level 24 marksStart

Definite integration and area under a curve

60 questions

LessonNot started
(a) Express

lim⁡δx→0∑x=3125x δx\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x

as an integral.

(1 mark)
(b) Hence show that

lim⁡δx→0∑x=3125x δx=ln⁡k\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x = \ln k

where kk is a constant to be found.

(2 marks)
●●●●●Level 23 marksStart

Areas with integration

60 questions

LessonNot started
The diagram shows the curve y=4x−x2y = 4x - x^2. The finite region RR, shown shaded, is enclosed by the curve and the xx-axis. Find the area of RR.123412345xy
●●●●●Level 23 marksStart
In this question you must show detailed reasoning.

Find the exact area of the region enclosed by the curve y=43x+2y = \dfrac{4}{3x + 2}, the two axes and the line x=2x = 2.

(3 marks)
●●●●●Level 23 marksStart

Integrating standard functions

39 questions

LessonNot started
Find ∫(e2x+cos⁡3x−4x)dx\displaystyle\int \left(e^{2x} + \cos 3x - \frac{4}{x}\right) dx.
●●●●●Level 23 marksStart
Find ∫(6e3x−sin⁡x)dx\displaystyle\int \left(6\mathrm{e}^{3x} - \sin x\right) dx. (3 marks)
●●●●●Level 23 marksStart
Find ∫32x+5 dx\displaystyle\int \dfrac{3}{2x + 5}\,dx. (3 marks)
●●●●●Level 23 marksStart
(a) Express

lim⁡δx→0∑x=3125x δx\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x

as an integral.

(1 mark)
(b) Hence show that

lim⁡δx→0∑x=3125x δx=ln⁡k\lim_{\delta x \to 0} \sum_{x = 3}^{12} \frac{5}{x}\,\delta x = \ln k

where kk is a constant to be found.

(2 marks)
●●●●●Level 23 marksStart
Find ∫12x−1 dx\displaystyle\int \frac{1}{2x - 1}\,dx.
●●●●●Level 22 marksStart
(a) Differentiate the following with respect to xx.

(i) ln⁡(3x2+5)\ln(3x^2 + 5)

(ii) xcos⁡4xx\cos 4x

(5 marks)
(b) Find ∫sin⁡3x dx\displaystyle\int \sin 3x\,dx.

(2 marks)
(c) The gradient of a curve is given by dydx=4e2x\dfrac{dy}{dx} = 4e^{2x} and the curve passes through the point (0,5)(0, 5).

Find the equation of the curve.

(2 marks)
●●●●●Level 29 marksStart
In this question you must show detailed reasoning.

Find the exact area of the region enclosed by the curve y=43x+2y = \dfrac{4}{3x + 2}, the two axes and the line x=2x = 2.

(3 marks)
●●●●●Level 23 marksStart
Find ∫(4x+3)7 dx\displaystyle\int (4x + 3)^7\,dx.
●●●●●Level 22 marksStart

Integrating with trigonometric identities

24 questions

LessonNot started
Find ∫cos⁡2x dx\displaystyle\int \cos^2 x\,dx.
●●●●●Level 22 marksStart
Find ∫sin⁡2x dx\displaystyle\int \sin^2 x\,dx.
●●●●●Level 22 marksStart

Integration by reversing the chain rule

24 questions

LessonNot started
Find ∫2x(x2+1)3 dx\displaystyle\int 2x(x^2 + 1)^3\,dx.
●●●●●Level 22 marksStart
Find ∫3x2(x3−2)5 dx\displaystyle\int 3x^{2}(x^{3} - 2)^{5}\,dx. (3 marks)
●●●●●Level 23 marksStart
Find ∫6x2x3+4 dx\displaystyle\int \frac{6x^2}{x^3 + 4}\,dx.
●●●●●Level 22 marksStart

Integration by substitution

36 questions

LessonNot started
Using the substitution u=x2−3u = x^2 - 3, find ∫x(x2−3)4 dx\displaystyle\int x(x^2 - 3)^4\,dx.
●●●●●Level 23 marksStart

Integration by parts

32 questions

LessonNot started
Find ∫xcos⁡x dx\displaystyle\int x\cos x\,dx.
●●●●●Level 23 marksStart
Find ∫x ex dx\displaystyle\int x\,e^x\,dx.
●●●●●Level 23 marksStart

Differential equations

56 questions

LessonNot started
A blacksmith takes a horseshoe out of a forge and places it on the stone floor of a workshop. The temperature of the workshop is a constant 15 ∘C15\,^\circ\text{C}

At time tt minutes after the horseshoe is taken out of the forge, the temperature of the horseshoe, θ ∘C\theta\,^\circ\text{C}, decreases at a rate which is proportional to θ−15\theta - 15

When the horseshoe is taken out of the forge, its temperature is 815 ∘C815\,^\circ\text{C} and its temperature is decreasing at a rate of 60 ∘C60\,^\circ\text{C} per minute.

Show that

dθdt=−0.075(θ−15)\frac{d\theta}{dt} = -0.075(\theta - 15)

(3 marks)
●●●●●Level 23 marksStart
Find the general solution of dydx=xy\dfrac{dy}{dx} = xy.
●●●●●Level 23 marksStart