Antiderivative

Q.

What is the formula of integration of $uv$?

Q. Integrate:$\int \frac{dt}{6t-1}$

1. $\frac{1}{6}ln(6t-1)+C$
2. $\frac{1}{12}ln(6t-1)+C$
3. $\frac{1}{6}ln(12t-1)+C$
4. $\frac{6}{1}ln(6t-1)+C$

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

1. $\ln |x|-\frac{x^3}{3}-\frac{1}{3}x+c$
2. $x-\frac{x^2}{3}-\frac{x^2}{3}+c$
3. $\frac{x^2}{2}-\frac{x^3}{3}+c$
4. $x-\frac{x^2}{2}-\frac{x^3}{3}+c$

Q. Evaluate: $\int (3e^{3x}+e^{-x})dx$

1. $e^{3x}+e^{-x}+C$
2. $3e^{3x}+e^{-x}+C$
3. $e^{3x}-e^{-x}+C$
4. $e^{3x}+e^{-2x}+C$

Q. Evaluate:
i. $\int (\sin x+\cos x)dx$
ii. $\int (3x^2+4)dx$

1. $i.-\cos x+\sin x+C$
   $ii.x^3+4x+C$
2. $i.-\cos x-\sin x+C$
   $ii.x^3-2x+C$
3. $i.-\cos x-\sin x+C$
   $ii.x^3+2x+C$
4. $i.\cos x+\sin x+C$
   $ii.x^3-4x+C$

Q. $\int \left(\frac{t\sqrt{t}+\sqrt{t}}{t^2}\right)dt$ is equal to

1. $\frac{1}{2\sqrt{2}}(\sqrt{t}-\frac{1}{\sqrt{t}})+c$
2. $2(\sqrt{t}-\frac{1}{\sqrt{t}})+c$
3. $2\sqrt{2}(\sqrt{t}-\frac{1}{\sqrt{t}})+c$
4. $\frac{1}{2}(\sqrt{t}-\frac{1}{\sqrt{t}})+c$

Q. Evaluate $\int \frac{3}{-3x+6}dx$

1. $\ln (-3x+6)$
   
2. $-\ln (-3x+6)$
   
3. $\frac{-\ln (-3x+6)}{3}$
   
4. $\frac{\ln (-3x+6)}{3}$

Q. Evaluate : $\int \frac{1}{{\sin }^{2}x{\cos }^{2}x}dx$

1. $(\tan x+\cot x+C)$
2. $(\tan x-\cot x+C)$
3. $(\tan x+2\cot x+C)$
4. $(\tan x+3\cot x+C)$

Q. Evaluate: $\int \left(\cos \right(x)+x^2)dx$

1. $\sin x+\frac{x^3}{2}+C$
2. $\sin x+\frac{x^3}{6}+C$
3. $\sin x+\frac{x^2}{3}+C$
4. $\sin x+\frac{x^3}{3}+C$

Q.

Evaluate ${\sec }^{-1}\left(\sec \right(-{30}^{°}\left)\right)$$=$

1. $-60°$

2. $-30°$

3. $30°$

4. $150°$

Q.

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left|\cos \frac{x}{2}\right|dx=$

1. $1$

2. $-2$

3. $\sqrt{2}$

4. $0$

Q. Find the integral $\int \frac{x+1}{x^2+2x-4}dx$

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

Q.

Integrate cos (4x+3) w.r.t x

1. $\frac{sin(4x+3)}{4}+c$

2. $-\frac{sin(4x+3)}{4}+c$

3. sin(4x+3)+c

4. -sin(4x+3)+c

Q. Evaluate $\int \cos (-2x+3)dx$

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

Q.

Find the integral of the given function w.r.t - x

$y=e^{2x}+\frac{1}{x^2}$

1. 

2. 

3. 

4. 

Q.

Find the integral of the given function w.r.t - x

$y=e^{2x}+\frac{1}{x^2}$

1. $\frac{2}{e^{2x}}-\frac{1}{x}+c$

2. $\frac{e^x}{2}-\frac{1}{x}+c$

3. $\frac{e^{2x}}{2}-\frac{1}{x}+c$

4. $e^{2x}-\frac{1}{x}+c$

Q.

Find the integral of the given function w.r.t x

$y=sin\left(8x\right)+x$

1. 

2. 

3. 

4. 

Q. Evaluate $\int (x^3-e^x+\cos x)dx$

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

Q. $\int (se{c}^{2}x+e^x+e^{3x})dx$ is equals to

1. $\sec x+\tan x+e^x+e^{3x}+C$
2. $\tan x+e^x+e^{3x}+C$
3. $\tan x+e^x+\frac{e^{3x}}{3}+C$
4. $\sec x+e^x+\frac{e^{3x}}{3}+C$

Q.

If $x^{2r}$ occurs in ${\left(x+\frac{2}{x^2}\right)}^n$, then $n-2r$ must be of the form:

1. $3k-1$

2. $3k$

3. $3k+1$

4. $3k+2$

Q. Evaluate: $\int {\sin }^{2}xdx$

1. $\frac{x}{2}-\frac{\sin 2x}{4}+C$
2. $\frac{x}{2}-\frac{\sin 2x}{2}+C$
3. $\frac{x}{2}-\frac{\sin 2x}{3}+C$
4. $\frac{x}{4}-\frac{\sin 2x}{4}+C$

Q. What is the antiderivative of $cosx$ ?

1. cos x
2. - sin x
3. tan x
4. sin x

Q. $\int 2x(x-x^{-3})dx$ is equal to

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

Q.

Evaluate: $\int \frac{2zdz}{\sqrt[3]{3z^2+1}}$

1. $\frac{3}{2}(z^2+1{)}^{-2/3}+c$

2. $\frac{3}{2}(z^2+1{)}^{2/3}+c$

3. $\frac{3}{2}(z{)}^{4/3}+c$

4. None of these

Q.

Can you use Sin Cos Tan on non right triangles?

Q. $\int (\frac{1}{x}-e^x-\frac{1}{3})dx$ is equal to

1. $\frac{1}{x^2}+e^x-\frac{1}{3}x+c$
2. $\ln |x|-e^x-\frac{1}{3}x+c$
3. $\ln |x|+e^x-\frac{1}{3}x+c$
4. $\frac{-1}{x^2}+e^x-\frac{1}{3}x+c$

Q. Evaluate $\int e^x\sin xdx$

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

Q.

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{dy}{dx}$ is equal to:

1. $\frac{1}{x^2{y}^2}$

2. $\frac{1}{x^2{y}^3}$

3. $\frac{1}{x^{3}y}$

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

Q. Change the order of the integration in
${\int }_0^a{\int }_0^{y}f\left(x,y\right)dxdy$

Q. $\int \left(\frac{t\sqrt{t}+\sqrt{t}}{t^2}\right)dt$ is equal to

1. $2(\sqrt{t}+2\sqrt{t})$
2. $2(\sqrt{t}+\frac{1}{\sqrt{t}})+c$
3. $2(\frac{1}{\sqrt{t}}-\sqrt{t})+c$
4. $2(\sqrt{t}-\frac{1}{\sqrt{t}})+c$