Integration by Parts

Q.

Find the value of ${\int }_0^{\infty }x{e}^{-x}dx$

1. 1

2. -1

3. 0

4. None of these

Q.

ILATE rule should always be applied whenever we are using integration by parts

1. True

2. False

Q. Lim x to pie ÷4 1-tanx÷x-pie÷4

Q.

Find the value of tan 3A - tan2A - tanA is equal to ________

1. tan 3A tan2A tanA

2. tan A tan2A - tan2A + tan3A - tan3AtanA

3. tan 3A + tan2A

4. -tan 3A tan 2A tanA

Q.

Let $A\{2,3,5,7\}.$ Examine whether the statements given below are true or false.

(i) $\exists x\in A$ such that $x+3>9.$

(ii) $\exists x\in A$ such that x is even.

(iii) $\exists x\in A$ such that $x+2=6.$

(iv) $\forall x\in A,x$ is prime.

(v) $\forall x\in A,x+2<10$

(vi) $\forall x\in A,x+4\ge 11$

Q. If $\int f\left(x\right)dx=\psi \left(x\right)\text{, then}\int x^{5}f\left(x^3\right)dx$ is equal to

1. $\frac{1}{3}\left[x^3\psi \right(x^3)-\int x^2\psi (x^3\left)dx\right]+c$
2. $\frac{1}{3}{x}^3\psi \left(x^3\right)-3\int x^3\psi \left(x^3\right)dx+c$
3. $\frac{1}{3}{x}^3\psi \left(x^3\right)-\int x^2\psi \left(x^3\right)dx+c$
4. $\frac{1}{3}\left[x^3\psi \right(x^3)-\int x^3\psi (x^3\left)dx\right]+c$

Q. Prove that integration of sec²x with respect to x is equal to tanx

Q.

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

1. $\frac{f\left(x\right)}{g^{\prime }\left(x\right)}$

2. $f^{\prime }\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)$

3. $f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$

4. $f\left(x\right)g^{\prime }\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)$

Q. If $I=\int \frac{dx}{x^3\sqrt{x^2-1}}$, then $I$ equals

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

Q. If $\Phi \left(x\right)={lim}_{n\to \infty }\frac{x^n-x^{-n}}{x^n+x^{-n}},0<x<1,ϵN$, then $\int \left(si{n}^{-1}x\right)\left(\Phi \right(x\left)\right)dx$ is equal to

1. $xsi{n}^{-1}\left(x\right)+\sqrt{1-x^2}+C$
2. $-\left(xsi{n}^{-1}\right(x)+\sqrt{1-x^2})$
3. $xsi{n}^{-1x+\sqrt{1-x^2}+C}$
4. None of these

Q. The integral $\int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx$ is equal to:

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

Q. Let $g\left(x\right)=\int \frac{1+2\cos x}{(\cos x+2{)}^2}dx$ and $g\left(0\right)=0$, the value of $8g\left(\frac{\pi }{2}\right)$ is

Q. For ${\sin }^{-1}x+{\cos }^{-1}y+{\sec }^{-1}z\ge t^2-\sqrt{2\pi }t+3\pi$, then the principal value of ${\cos }^{-1}\left(\cos 5t^2\right)$ is,

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

Q.

In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
$\int \left[f\right(x\left)h\right(x)+g(x\left)\right]dx$
Let $\int f\left(x\right)h\left(x\right)dx=I_1$ and $\int g\left(x\right)dx=I_2$
Integrating $I_1$ by parts, we get
$I_1=f\left(x\right)\int h\left(x\right)dx-\int \left\{f^{\prime }\right(x)\int h(x\left)dx\right\}dx$

$\int \frac{x{e}^x}{(1+x{)}^2}dx$ is equal to

1. $x{e}^x+c$

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

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

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

Q. If $\cos x=\tan y$; $\cos y=\tan z$; $\cos z=\tan x$ then the value of $\sin x$ is

1. $\cos {18}^0$
2. $2\cos {18}^0$
3. $\sin {18}^0$
4. $2\sin {18}^0$

Q. The integral $\int \cos \left({\log }_{e}x\right)dx$ is equal to: (where $C$ is a constant of integration)

1. $x\left[\cos \right({\log }_{e}x)+\sin ({\log }_{e}x\left)\right]+C$
2. $x\left[\cos \right({\log }_{e}x)-\sin ({\log }_{e}x\left)\right]+C$
3. $\frac{x}{2}\left[\sin \right({\log }_{e}x)-\sin ({\log }_{e}x\left)\right]+C$
4. $\frac{x}{2}\left[\cos \right({\log }_{e}x)+\sin ({\log }_{e}x\left)\right]+C$

Q. The value of ${\int }_0^1(x^3+3e^x+4)(x^2+e^x)dx$ is -

1. $(3e+2)/6$
2. $(3e-2)/6$
3. { ${(3e+2)}^2/36$ }
4. None of these.

Q. $\int \frac{(3\sin \varphi -2)\cos \varphi }{(5-{\cos }^2\varphi -4\sin \varphi )}d\varphi$ is

1. $3\ln (2-\sin \varphi )\frac{4}{(\sin \varphi -2)}+c$
2. $3\ln (\sin \varphi -2)+\frac{4}{2-\sin \varphi }+c$
3. $\ln (2-\sin \varphi )+\frac{4}{2-\sin \varphi }+c$
4. $3\ln (2-\sin \varphi )+\frac{4}{(2-\sin \varphi )}+c$

Q. 5.A = cos6x + 6 cos4x + 15cos2x + 10

Q. If $\int {\sin }^{-1}x{\cos }^{-1}xdx=f^{-1}\left(x\right)[Ax-x{f}^{-1}(x)-2\sqrt{1-x^2}]+\frac{\pi }{2}\sqrt{1-x^2}+2x+c,$ then

1. $f\left(x\right)=\sin x$
2. $f\left(x\right)=\cos x$
3. $A=\frac{\pi }{4}$
4. $A=\frac{\pi }{2}$

Q. $\int e^{-3x}\sin 4xdx=$

1. $\frac{e^{-3x}(4\cos 4x-3\sin 4x)}{25}+c$
2. $\frac{-e^{-3x}(3\cos 4x+4\sin 4x)}{25}+c$
3. $\frac{-e^{-3x}(4\cos 4x+3\sin 4x)}{25}+c$
4. None of these

Q. Solve this:

Q. If $\int e^{ax}cos\left(bx\right)dx=\frac{e^{ax}}{K}(acosbx+bsinbx)+C$, then the K here would be equal to -

1. $\sqrt{a^2+b^2}$
2. $(a^2+b^2{)}^{\frac{1}{3}}$
3. $a^2+b^2$
4. $(a^2-b^2)$

Q. $\mathrm{The}\int \left(\frac{\mathrm{cosx}}{x}-\log x^{\mathrm{sinx}}\right)\mathrm{dx}\mathrm{is}\mathrm{equal}\mathrm{to}$.

1. $\mathrm{cosx}\mathrm{logx}+C$
2. $\mathrm{sinx}\mathrm{logx}+C$
3. $\frac{\mathrm{cosx}}{\log x}+C$