• A Level Maths

Demo

• A Level
• Integration (Basic)

1. Find $$\int 4x-1\ \mathrm{d}x$$

$2x^2-x+c$

2. Find $$\int 3u^2-4u-3\ \mathrm{d}u$$

$u^3-2u^2-3u+c$

3. Find $$\int 5+3r^2-4r^3\ \mathrm{d}r$$

$5r+r^3-r^4+c$

4. Find $$\int \theta^4+6\theta^2-\theta\ \mathrm{d}\theta$$

$\dfrac{1}{5}\theta^5 + 2\theta^3 - \dfrac{1}{2}\theta^2 + c$

5. Find the area under the curve $$y=3+\dfrac{5}{x^2}$$ between $x=-5$ and $x=-1$.

$16$

6. Find the area under the curve $$y=2x+3x^{\frac{1}{2}}$$ between $x=1$ and $x=4$.

$29$

7. Find $a$ given $$\displaystyle\int_1^4 3x^2+ax-5\ \mathrm{d}x=18$$

$-4$

8. Find the area bound by the curve $y = x\sqrt{x} - 3x$, the $x$ axis, and the line $x = a$, where $a$ is the $x$ coordinate of the turning point of the curve.

$11.2$

9. Find $$\int 3\sin^2\theta + 3\cos^2\theta\ \mathrm{d}x$$

$3x+c$

10.

1. Find the coordinates where the lines $y=x^2-3x+4$ and $y=x+1$ intersect.
2. Hence find the area enclosed by these curves.

1. $(1,2)$ and $(3,4)$
2. $\dfrac{4}{3}$

11. Find the area enclosed by the curves $y=\sqrt{x}$ and $x-2y=0$.

$\dfrac{4}{3}$

12. Evaluate $$\int_1^{\infty}\frac{\sqrt{x}(3-\sqrt{x})}{x^4}\ \mathrm{d}x$$

$\dfrac{7}{10}$

13. Evaluate $$\int_1^{\infty}\frac{(x^{-\frac{3}{4}}-\sqrt[3]{x})^2}{2x^2}\ \mathrm{d}x$$

$\dfrac{169}{170}$

14.

1. Estimate the area under the graph of $y = (2x+1)^7$ between $x = 1$ and $x = 3$ by using the first three terms in ascending powers of $x$ in the binomial expansion.
2. The actual area is in the region of 350,000. Explain why your estimate is so poor.

1. $786$
2. Using the first three terms is only valid for small values of $x$.

15.

1. Assuming that $\theta$ is small and in radians, estimate $$\int_0^1\dfrac{3-2\cos\theta}{\sqrt{\tan\theta}}\ \mathrm{d}\theta$$
2. The real value of this is approximately $2.27$. Explain whether your estimate was a good estimate.

1. $2.40$
2. Not very good because the upper limit of the integral is too large for the approximation.

16.

1. Evaluate $\displaystyle\int_1^2\frac{8}{x^3}\ \mathrm{d}x$
2. Hence find the area bound by the curve $y = \dfrac{8}{x^3}$, the $y$ axis, and the lines $y = 1$ and $y = 8$.

1. $3$
2. $9$

17. The diagram below shows two curves: $y = x(x-4)$ and $y = 4\sqrt{x}-x\sqrt{x}$. Find the area bound by the two curves.

$19.2$

18. Find the area bound by the two curves $y = 2x\sqrt{x}$ and $y = 8\sqrt{x}$.

$\dfrac{256}{15}$