# Techniques of Integration

###### 5 Essential Questions

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

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

##### Suggested Handwritten and Video Solutions ##### ACJC Tutorial 21 Q15

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

##### Suggested Handwritten and Video Solutions  ##### 2020 EJC P1 Q2

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



##### Suggested Handwritten and Video Solutions ##### 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).



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



(ii)

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



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



(b)

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



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



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• (b)     ##### 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

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



(i) By considering the shaded rectangle, show that

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



(ii)

Deduce that

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



(iv) Deduce that

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



(v)

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

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



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

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



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