Binomial Integral

Q.

$\int \frac{\sqrt{1+x^2}}{x^4}dx$

Q.

$\int \frac{\sqrt{x^2+1}\left[log\right(x^2+1)-2logx]}{x^4}dx$

Q. Find the following integrals:
$\left(i\right)\int \frac{(x^3-1)}{x^2}dx$
$\left(ii\right)\int (x^{\frac{2}{3}}+1)dx$
$\left(iii\right)\int (x^{\frac{3}{2}}+2e^x-\frac{1}{x})dx$

Q. Find the integral: $\int \frac{x^3-x^2+x-1}{x-1}dx$

Q. The value of the integral $\int \frac{e^{x^2+4\ln x}-x^3{e}^{x^2}}{x-1}dx$ equal to
(where $C$ is the constant of integration)

1. $\frac{(x^2-1)e^{x^2}}{2}+C$
2. $\frac{(x-1)x{e}^{x^2}}{2}+C$
3. $\frac{(x^2-1)e^{x^2}}{2x}+C$
4. $\frac{(x+1)e^{x^2}}{2x}+C$

Q. Find the integral: $\int (2x^2+e^x)dx$

Q. Consider $I_n=\underset{0}{\overset{\infty }{\int }}\frac{dx}{{(x+\sqrt{1+x^2})}^n},$ where $n>1$, then which of the following statement(s) is/are true ?

1. $I_2=\frac{2}{3}$
2. $I_2=\frac{2}{5}$
3. $I_{10}<I_5$
4. $I_{10}>I_5$

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

1. $\frac{2}{\sqrt{3}}\left(ta{n}^{-1}\right(\frac{\sqrt{x+1}}{\sqrt{3}}\left)\right)+C$
2. $\frac{2}{\sqrt{3}}\left(ta{n}^{-1}\right(\frac{\sqrt{x-2}}{\sqrt{3}}\left)\right)+C$
3. $\frac{2}{\sqrt{3}}\left(ta{n}^{-1}\right(\frac{\sqrt{x-2}}{\sqrt{2}}\left)\right)+C$
4. $\frac{1}{\sqrt{2}}\left(ta{n}^{-1}\right(\frac{\sqrt{x-2}}{\sqrt{2}}\left)\right)+C$

Q. If $\int \frac{dx}{x\sqrt{1-x^3}}=alog\left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right|+C$ then a =

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

Q. $\int \frac{\sqrt{1+\sqrt[3]{3x}}}{\sqrt[3]{3x^2}}dx$ is equal to

1. $-(1+x^{1/3}{)}^{3/2}+C$
2. $2(1+x^{1/3}{)}^{3/2}+C$
3. $(1+x^{1/3}{)}^{3/2}+C$
4. None of these

Q. $\int \frac{x^2-2}{x^3\sqrt{x^2-1}}dx$ is equal to

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

Q. Prove that: $[1+\frac{1}{{\tan }^2\theta }][1+\frac{1}{{\cot }^2\theta }]=\frac{1}{{\sin }^2\theta -{\sin }^4\theta }$

Q. $\int \frac{\sqrt{1+\sqrt[3]{3x}}}{\sqrt[3]{3x^2}}dx$ is equal to

1. $(1+x^{1/3}{)}^{3/2}+C$
2. $-(1+x^{1/3}{)}^{3/2}+C$
3. $2(1+x^{1/3}{)}^{3/2}+C$
4. None of these

Q. $\int \frac{dx}{(x+1)\sqrt{x^2-1}}=$

1. $\sqrt{\frac{x+1}{x-1}}$
2. $\sqrt{\frac{x-1}{x+1}}$
3. $\sqrt{\frac{x+1}{x^2+1}}$
4. $\sqrt{\frac{x^2+1}{x-1}}$

Q. $\int \left(x{)}^{\frac{1}{3}}\right(2+x^{\frac{1}{2}}{)}^{2}dx$

1. 
2. 
3. 
4. 

Q. ${\int }_1^2\sqrt{(x-1)(2-x)}dx=$

1. $\frac{\pi }{8}$
2. $\frac{\pi }{4}$
3. $\frac{1}{8}$
4. none

Q. The value of the integral $\int \frac{x}{(x-1{)}^2(x+2)}dx$
(where $m$ is an arbitrary constant)

1. $\frac{2}{9}\ln \left|\frac{x-2}{x+4}\right|+\frac{1}{3(x-1)}+m$
2. $\frac{2}{9}\ln \left|\frac{x-1}{x+2}\right|-\frac{1}{3(x-1)}+m$
3. $\frac{2}{3}\ln \left|\frac{x+1}{x+2}\right|-\frac{1}{3(x-1)}+m$
4. $\frac{2}{3}\ln \left|\frac{x-1}{x+3}\right|-\frac{1}{3(x-1)}+m$

Q. $\int \frac{1}{(x+1)\sqrt{x-2}}dx$

1. 
2. 
3. 
4. 

Q. $\text{The integral}\int \frac{dx}{(x+1{)}^{\frac{3}{4}}(x-2{)}^{\frac{5}{4}}}\text{is equal to}$

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

Q. $\int \left(x{)}^{\frac{1}{3}}\right(2+x^{\frac{1}{2}}{)}^{2}dx$

1. $7x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
2. $4x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
3. $3x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
4. $11x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$

Q. $\text{The integral}\int \frac{dx}{(x+1{)}^{\frac{3}{4}}(x-2{)}^{\frac{5}{4}}}\text{is equal to}$

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

Q. $\text{The integral}\int \frac{dx}{(x+1{)}^{\frac{3}{4}}(x-2{)}^{\frac{5}{4}}}\text{is equal to}$

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