AS-Level Maths | Integration
To integrate: raise the power by one, then divide the coefficient by the new power.
$\displaystyle\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C$
Using limits gives the area between the curve and the $x$-axis.
Question 1
Integrate the following expressions with respect to $x$.
a) $\displaystyle\int 2\sqrt{x} - \frac{1}{x^2} \, dx$
b) $\displaystyle\int 4\sqrt{x} - 2\sqrt{x^5} \, dx$
c) $\displaystyle\int \frac{3}{4\sqrt{x}} + \frac{1}{x^3} \, dx$
d) $\displaystyle\int 2x\sqrt{x} - \frac{4}{3x^2} \, dx$
e) $\displaystyle\int 2\sqrt{x} + \frac{1}{2\sqrt{x}} \, dx$
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a) $\dfrac{4}{3}x^{\frac{3}{2}} + \dfrac{1}{x} + C$
b) $\dfrac{8}{3}x^{\frac{3}{2}} - \dfrac{4}{7}x^{\frac{7}{2}} + C$
c) $\dfrac{3}{2}\sqrt{x} - \dfrac{1}{2x^2} + C$
d) $\dfrac{4}{5}x^{\frac{5}{2}} + \dfrac{4}{3}x^{-1} + C$
e) $\dfrac{4}{3}x^{\frac{3}{2}} + \sqrt{x} + C$
Question 2
The point $P(8,18)$ lies on the curve $C$, whose gradient function is given by $\displaystyle\frac{dy}{dx} = 8\sqrt[3]{x} - 10, \; x \geq 0$.
Find an equation for $C$.
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$y = 6x^{\frac{4}{3}} - 10x + 2$
Question 3
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a) $(-5, 0)$ and $(3, 0)$
b) $-27$
Question 4
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a) $(0,0)$, $(2,0)$ and $(3,0)$
b) Total area $= \dfrac{8}{3} + \dfrac{5}{12} = 3\dfrac{1}{12}$
Question 5
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a) $(1, 2)$ and $(3, 4)$
b) Area $= \dfrac{4}{3}$
Question 6
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a) $3$
b) Area $= 9$
Question 7
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a) $y = 2 - x$
b) Area $= \dfrac{5}{6}$
Question 8
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a) $y = 4 - x$
b) Area $= \dfrac{5}{3}$
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