Theorems on Integration

Q.

$\frac{d}{dx}\left(\frac{secx+\tan x}{secx–\tan x}\right)=?$

1. $\frac{2\cos x}{{(1-\sin x)}^2}$

2. $\frac{\cos x}{{(1-\sin x)}^2}$

3. $\frac{2\cos x}{(1-\sin x)}$

4. None of these

Q. $\int x{e}^{x^2\log 2}{e}^{x^2}dx=$ ____ $+c$

1. $\frac{(2e{)}^{x^2}}{\log \left(2e\right)}$
2. $\frac{2^{x^2}{e}^{x^2}}{1+\log 2}$
3. $\frac{2^{x^2}{e}^{x^2}}{2(1+\log 2)}$
4. $\frac{2^{x^2\log 2}{e}^{x^2}}{\log 2}$

Q. For each positive integer $n$, let
$y_n=\frac{1}{n}\left\{\right(n+1\left)\right(n+2)......(n+n){\}}^{\frac{1}{n}}$

For $x\in R$ let $\left[x\right]$ be the greatest integer less than or equal to $x$. If $\underset{n\to \infty }{lim}{y}_n=L$, then the value of $\left[L\right]$ is

Q.

If $y=\log \left[\sin \left(x^2\right)\right],0<x<\frac{\pi }{2}$, then $\frac{dy}{dx}$at $x=\frac{\sqrt{\pi }}{2}$ is

1. $0$

2. $1$

3. $\frac{\mathrm{\pi }}{4}$

4. $\sqrt{\mathrm{\pi }}$

Q. Evaluate: $\int \frac{{\text{x}}^2}{({\text{x}}^2+{\text{a}}^2)({\text{x}}^2+{\text{b}}^2)}\text{dx}$

Q.

If $lo{g}_{x}a,a^{\frac{x}{2}}$ and $lo{g}_{b}x$ are in G.P. then write the value of x.

Q.

Integrate the function.
$\int e^x(\frac{1}{x}-\frac{1}{x^2})dx.$

Q. If $x=\text{cosec}\left({\tan }^{-1}\right(\cos \left({\cot }^{-1}\right(\sec \left({\sin }^{-1}a\right)\left)\right)\left)\right)$ and $y=\sec \left({\cot }^{-1}\right(\sin \left({\tan }^{-1}\right(\text{cosec}\left({\cos }^{-1}a\right)\left)\right)\left)\right),$ where $a\in [0,1],$ then which of the following options is(are) CORRECT?

1. $x=\sqrt{2-a^2}$
2. $x=\sqrt{3-a^2}$
3. $y=\frac{1}{\sqrt{2-a^2}}$
4. $y=\sqrt{3-a^2}$

Q.

Find the following integrals.
$\int \frac{x^3+5x^2-4}{x^2}dx.$

Q. The value of $\int \frac{e^{6\log x}-e^{5\log x}}{e^{4\log x}-e^{3\log x}}dx$ is equal to

1. $0$
2. $\frac{x^3}{3}$
3. $\frac{3}{x^3}$
4. $\frac{1}{x}$

Q.

Find the following integrals.
$\int \frac{x^3-x^2+x-1}{(x-1)}dx.$

Q. Integrate the function: $\frac{5x+3}{\sqrt{x^2+4x+10}}$

Q. (i) Find the integral: $\int \frac{dx}{x^2-16}$

(ii) Find the integral: $\int \frac{dx}{\sqrt{2x-x^2}}$

Q. Integrating factor of $x\frac{dy}{dx}-y=x^4-3x$ is

1. $\log x$
2. $x$
3. $\frac{1}{x}$
4. $-x$

Q. $\int \frac{\cos 2x-\cos 2\theta }{\cos x-\cos \theta }dx$ is equal to

1. $2(\sin x+x\cos \theta )+C$
2. $2(\sin x-x\cos \theta )+C$
3. $2(\sin x-2x\cos \theta )+C$
4. $2(\sin x+2x\cos \theta )+C$

Q.

Integrate the following functions.
$\int \frac{1}{\sqrt{x^2+2x+2}}dx.$

Q. If $3{\tan }^{-1}x+{\cot }^{-1}x=\pi$ then $x$ equal to

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

Q. The area cut off by the parabola $y^2=4ax$ and its latus rectum is

1. $8a^2$
2. $4a^2$
3. $\frac{8a^2}{3}$
4. $\frac{4a^2}{3}$

Q. 1)Find the maximum and minimum values, if any, of the function given by
$f\left(x\right)=(2x-1{)}^2+3$

2)Find the maximum and minimum values, if any, of the function given by
$f\left(x\right)=9x^2+12x+2$

3)Find the maximum and minimum values, if any, of the function given by
$f\left(x\right)=-(x-1{)}^2+10$

4)Find the maximum and minimum values, if any, of the function given by
$f\left(x\right)=x^3+1$

Q.

Evaluate.

Q.

$\int \frac{dx}{\sqrt{(1-x)(x-2)}}=$

1. ${\sin }^{-1}(2x-3)+c$

2. ${\sin }^{-1}(2x+5)+C$

3. ${\sin }^{-1}(3–2x)+C$

4. ${\sin }^{-1}(5–2x)+C$

Q.

Integrate the function.
$\int si{n}^{-1}\left(\frac{2x}{1+x^2}\right)dx.$

Q. Given that $\frac{dy}{dx}=e^{-2y}$ and $y=0$ when $x=5$.Find the value of $x$ when $y=3$.

Q. Find: $\int \frac{x^2+1}{x^2-5x+6}$

Q.

${\int }_3^{10}\left[ln\right[x\left]\right]dx$= ___, where [.] is GIF.

Q. If $x+\frac{1}{x}=3$, find the value of $x^4+\frac{1}{x^4}$.

Q. Integrate the function: $\frac{(1+\log x{)}^2}{x}$

Q.

Evaluate the definite integrals.
${\int }_1^2(4x^3-5x^2+6x+9)dx$

Q.

If $y=\left(\frac{3^x-1}{3^x+1}\right)\sin x+{\log }_e\left(1+x\right),x>-1$, then at $x=0$, $\frac{dy}{dx}$ is equal to

1. $1$

2. $0$

3. $-1$

4. $-2$

Q. Evaluate: $\int \frac{{\text{x}}^2}{{\text{x}}^4-{\text{x}}^2-12}\text{dx}$