Integration Using Substitution

Q. Solve the following equations:
(i) $\tan x+\tan 2x+\tan 3x=0$
(ii) $\tan x+\tan 2x=\tan 3x$
(iii) $\tan 3x+\tan x=2\tan 2x$

Q.

By introducing a new variable t, putting x=cos t , the expression is $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y$transformed into :

1. 

2. 

3. 

4. 

Q. $\tan A+\tan \left(60°+A\right)-\tan \left(60°-A\right)=3\tan 3A$

Q. If $\tan \frac{x}{2}=\frac{\sqrt{1-e}}{1+e}\tan \frac{\alpha }{2}$, then $\cos \alpha =$

(a) $1-e\cos \left(\cos x+e\right)$

(b) $\frac{1+e\cos x}{\cos x-e}$

(c) $\frac{1-e\cos x}{\cos x-e}$

(d) $\frac{\cos x-e}{1-e\cos x}$

Q.

What is the value of tan 25^o?

1. tan 30^o – tan 5^o

2. tan 20^o + tan 5^o

3. 

4. 

Q. Solve this:
Question 16
Prove that
(i) tan 3A tan 2A tan A = tan 3A - tan 2A - tan A
(ii) cot A cot 2A - cot 2A cot 3A - cot 3A cot A = 1

Q. If $\cos \theta =\frac{\cos \alpha +\cos \beta }{1+\cos \alpha \cos \beta }$, prove that $\tan \frac{\theta }{2}=\pm \tan \frac{\alpha }{2}\tan \frac{\beta }{2}$

Q. $\tan \mathrm{\theta }\tan \left(\mathrm{\theta }+60°\right)+\tan \mathrm{\theta }\tan \left(\mathrm{\theta }-60°\right)+\tan \left(\mathrm{\theta }+60°\right)\tan \left(\mathrm{\theta }-60°\right)=-3$

Q. $\underset{\frac{-\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}{\log }_e(\sin x+\cos x)dx$ is

1. $-\frac{\pi }{4}\ln 2$
2. $\frac{\pi }{4}\ln 2$
3. $\frac{\pi }{8}\ln 2$
4. $-\frac{\pi }{8}\ln 2$

Q. If $\mathrm{cosec}x-\cot x=\frac{1}{2},0<x<\frac{\mathrm{\pi }}{2},$, then cos x is equal to
(a) $\frac{5}{3}$

(b) $\frac{3}{5}$

(c) $-\frac{3}{5}$

(d) $-\frac{5}{3}$

Q. If $I_n={\int }_0^{\frac{\pi }{4}}{\tan }^{n}x\mathrm{dx}$, then the value of $I_9+I_{11}$ is

1. $\frac{1}{9}$
2. $\frac{1}{10}$
3. $\frac{1}{11}$
4. $\frac{1}{8}$

Q. $\underset{x\to 0^{+}}{\lim }{\left\{1+{\tan }^2\sqrt{x}\right\}}^{1/2x}$

Q. Find tan75, hence evaluate tan75 + cot 75

Q. $\underset{x\to 0}{\lim }\frac{5x\cos x+3\sin x}{3x^2+\tan x}$

Q. tan x +cot x = 2 then show that 1)tanⁿ x +cotⁿ x =2
2)tan^1/2 x+ cot^1/2x=2

Q. The integral $\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}{e}^x\left(\ln \right(\cos x\left)\right)(\cos x\sin x-1){\text{cosec}}^{2}xdx$ is equal to

1. $\frac{e^{\pi /4}}{2}[2e^{\pi /4}+\ln 2-2]$
2. $\frac{e^{\pi /4}}{2}[e^{\pi /4}+\ln 2-2]$
3. $\frac{e^{\pi /4}}{2}[e^{\pi /4}-\ln 2+2]$
4. $\frac{e^{\pi /4}}{2}[2e^{\pi /4}-\ln 2+2]$

Q. If $\underset{0}{\overset{\infty }{\int }}\frac{dx}{x^{3/2}+1}=\frac{a\pi }{b^{3/2}},$ where $\text{gcd}(a,b)=1,$ then

1. $a=2$
2. $b=3$
3. $a=4$
4. $b=5$

Q. Solve this:
Q.6. Show that :
tan 50$°$ = tan 40$°$ + 2 tan 10$°$

Q. If α and β are acute angles satisfying $\cos 2\alpha =\frac{3\cos 2\beta -1}{3-\cos 2\beta }$, then tan α =
(a) $\sqrt{2}\tan \beta$

(b) $\frac{1}{\sqrt{2}}\tan \beta$

(c) $\sqrt{2}\cot \beta$

(d) $\frac{1}{\sqrt{2}}\cot \beta$

Q. If $y=\underset{x^2}{\overset{x^3}{\int }}\frac{1}{\ln t}dt$ $(\text{where}x\in R^{+}-\{1\left\}\right)$, then the value of $\frac{dy}{dx}$ is

1. $x(x-1)(\ln x{)}^{-1}$
2. $x(x-1)(\ln x{)}^{-2}$
3. $x(x-1)(\ln x{)}^2$
4. $x(x-1)\left(\ln x\right)$

Q. If 0<$\theta$<$\frac{\mathrm{\pi }}{2}$and sin $\theta$ + cos ​$\theta$ +tan ​$\theta$+cot ​$\theta$ +sec ​$\theta$ +cosec ​$\theta$= 7, then show that sin 2​$\theta$ is a root of the equation $x^2$-44x+36 =0

Q. Find the general solutions of the following equations:
(i) $\sin 2\mathrm{\theta }=\frac{\sqrt{3}}{2}$
(ii) $\cos 3\mathrm{\theta }=\frac{1}{2}$
(iii) $\sin 9\mathrm{\theta }=\sin \mathrm{\theta }$
(iv) $\sin 2\mathrm{\theta }=\cos 3\mathrm{\theta }$
(v) $\tan \mathrm{\theta }+\cot 2\mathrm{\theta }=0$
(vi) $\tan 3\mathrm{\theta }=\cot \mathrm{\theta }$
(vii) $\tan 2\mathrm{\theta }\tan \mathrm{\theta }=1$
(viii) $\tan m\mathrm{\theta }+cotn\mathrm{\theta }=0$
(ix) $\tan p\mathrm{\theta }=cotq\mathrm{\theta }$
(x) $\sin 2\mathrm{\theta }+\cos \mathrm{\theta }=0$
(xi) $\sin \mathrm{\theta }=\tan \mathrm{\theta }$
(xii) $\sin 3\mathrm{\theta }+\cos 2\mathrm{\theta }=0$

Q. The value of ${\int }_0^{\pi }\frac{1}{5+4\cos x}\mathrm{dx}$ is

1. $\pi$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. $2{\cos }^{-1}x={\sin }^{-1}\left(2x\sqrt{1-x^2}\right)$ is valid for all values of $x$ satisfying

1. $-1\le x\le 1$
2. $0\le x\le 1$
3. $\frac{1}{\sqrt{2}}\le x\le 1$
4. $0\le x\le \frac{1}{\sqrt{2}}$

Q. From $1970$ to $1980$, the rate of potato consumption in a particular country was $C\left(t\right)=2.2+{1.1}^t$ millions of bushels per year, with $t$ being years since the beginning of $1970.$ How many bushels were consumed from the beginning of $1972$ to the end of $1973$?

1. $7.066$ million
2. $4.6$ million
3. $\mathrm{6..2}$ million
4. $6.067$ million