# Techniques of Integration

Integration is a fundamental topic in Mathematics that studies the relationship between a whole and its parts. Integrals are used in arithmetic, science and engineering, especially where it is necessary to study the dynamics of variable quantities like temperature or pressure. This is an operation we use all the time, to find areas under graphs, the volume of a solid, and many others.

##### ACJC Tutorial 21 – Integration II [Self Practice Questions] Q5

$\int{x\,\sin }\left( \frac{x}{2} \right)\,\text{d}x$

##### ACJC Tutorial 21 Q15

Evaluate $\int{\frac{1}{{{(1+{{x}^{2}})}^{2}}}\,\text{d}x}\,\,\,\,(\text{let}\,\,x=\,\tan \,\theta )$

##### 2020 EJC P1 Q2

State the derivative of $\tan {{x}^{2}}$. Hence, or otherwise, find $\int{{{x}^{3}}{{\sec }^{2}}{{x}^{2}}\text{d}x}$

[4]

##### 2020 NYJC J2 CT P1 Q6

(a)

(i)

Show that $\tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta }$ using the addition formula from the List of Formulae (MF26).

[1]

(i) Show that $\tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta }$ using the addition formula from the List of Formulae (MF26).

[1]

(ii)

By using the substitution $x=\tan \theta$, or otherwise, find the exact value of  $\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)}\,\text{d}x$.

[5]

(ii) By using the substitution $x=\tan \theta$, or otherwise, find the exact value of  $\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)}\,\text{d}x$.

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(b)

Find the exact value of $\int_{-1}^{1}{\frac{2+\left| x \right|}{2+x}\text{d}x}$.

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(b) Find the exact value of $\int_{-1}^{1}{\frac{2+\left| x \right|}{2+x}\text{d}x}$.

[4]

##### 2014 AJC P1 Q9

The graph of $y=\sqrt{x+40}$ is shown in the diagram below.

(i)

By considering the shaded rectangle, show that

$\sqrt{n+40}<\int\limits_{n}^{n+1}{\sqrt{x+40}\text{ d}x}$.

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(i) By considering the shaded rectangle, show that

$\sqrt{n+40}<\int\limits_{n}^{n+1}{\sqrt{x+40}\text{ d}x}$.

[1]

(ii)

Deduce that

$\sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{80}<$$\int\limits_{-40}^{41}{\sqrt{x+40}\text{ d}x}. [2] (ii) Deduce that \sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{80}<\int\limits_{-40}^{41}{\sqrt{x+40}\text{ d}x}. [2] (iii) Show also that \sqrt{n+41}>\int\limits_{n}^{n+1}{\sqrt{x+40}\text{ d}x}. [2] (iii) Show also that \sqrt{n+41}>\int\limits_{n}^{n+1}{\sqrt{x+40}\text{ d}x}. [2] (iv) Deduce that \sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{81}>$$\int\limits_{-40}^{41}{\sqrt{x+40}\text{ d}x}$.

[1]

(iv) Deduce that

$\sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{81}>\int\limits_{-40}^{41}{\sqrt{x+40}\text{ d}x}$.

[1]

(v)

Hence, deduce that the value $a$, where $a\in \mathbb{Z}$, that satisfies the following inequality,

$9a<\sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{81}<9\left( a+1 \right)$.

[2]

(v) Hence, deduce that the value $a$, where $a\in \mathbb{Z}$, that satisfies the following inequality,

$9a<\sqrt{1}+\sqrt{2}+\sqrt{3}+…+\sqrt{81}<9\left( a+1 \right)$.

[2]

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