Integration as Antiderivative

Q. 19. if 2sinA/1+cosA+sinA=y then prove that 1-cosA+sinA/1+sinA is also y

Q. 52. prove that sin(x+y)=sinx.cosy+cosx.siny

Q. 45. TRIGONOMETRIC EQUATIONS: If sin²4x + cos²x = 2 sin4x cos²x, then x = A. n(pie) /2 B. (2n+1)pie/2 C. (2n+1)pie D. None of these

Q. 45 If sinx + siny =1/2 and cosx +cosy = 1, then tan(x+y) =

Q. 38.4sinx*sin2x*sin4x=sin3x Find the value if x?

Q.

Describe Methods of Integration

Q. 40. P.t sin (x+y)÷sin (x-y)=tanx+tan y÷tanx-tan y

Q. Find the derivative of the following function:
f(x)= $x^4(5sinx-3cosx)$

Q. Find the derivative of tan2x using first principle

Q.

cos130° cos40°+ sin130° sin40° is

1. 1

2. 0

3. 1/2

4. Not defined

Q. Given y= sin t^3 dy/dt=d[(sin t)^3]/dt dy/dt=3(sin t)^2 × d(sin t)/dt dy/dt=3(sin t)^2 × cos t explanation needed for step 2

Q. Let $f:[0,1]\to R$ be such that $f\left(xy\right)=f\left(x\right)\cdot f\left(y\right)$, for all $x,y\in [0,1]$, and $f\left(0\right)\ne 0$. If $y=y\left(x\right)$ satisfies the differential equation, $\frac{dy}{dx}=f\left(x\right)$ with $y\left(0\right)=1,\text{then}y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to :

1. $2$
2. $3$
3. $4$
4. $5$

Q. Let g(x) be an antiderivative for f(x). Then In $(1+(g\left(x\right){)}^2)$ is an antiderivate for

1. $\frac{2f\left(x\right)g\left(x\right)}{1+\left(f\right(x){)}^2}$
2. $\frac{2f\left(x\right)g\left(x\right)}{1+\left(g\right(x){)}^2}$
3. $\frac{2f\left(x\right)}{1+\left(f\right(x){)}^2}$
4. None

Q.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$ are two real function. Find $f+g\left(x\right)$

Q. 52.Prove that cos2xcosx/2-cos3x cos 9x/2= sin 5xsin 5x/2

Q. A continuous and differentiable function having an unique derivative will always have an unique anti-derivative.

1. False
2. True

Q. If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b.

Q.

If $\int f\left(x\right)dx=F\left(x\right)$, then $\int 5\times f\left(x\right)dx=\frac{F\left(x\right)}{5}$, because integration is the inverse process of differentiation

1. True

2. False

Q. 28. What is the curve represented by x=3(cos t +sin t) and y= 4(cos t- sin t) ?

Q. $\text{Let}\lambda ={\int }_1^9(\frac{1}{2}+\sqrt{\frac{1}{4}+{\log }_{3}x})dx+{\int }_1^2\frac{3^{x^2-\left[x\right]}}{3^{\left\{x\right\}}}dx$ then $2\lambda \text{equals}$ [where [.] denotes the greatest integer function and {.} denotes the fractional part function]

Q. Let $\int e^x\left\{f\right(x)-f^{\prime }(x\left)\right\}dx=\varphi \left(x\right)$. Then $\int e^{x}f\left(x\right)dx$ is
(where $c$ is constant of integration)

1. $\varphi \left(x\right)-e^{x}f\left(x\right)+c$
2. $\varphi \left(x\right)+e^{x}f\left(x\right)+c$
3. $\frac{1}{2}\left\{\varphi \right(x)+e^{x}f(x\left)\right\}+c$
4. $\frac{1}{2}\left\{\varphi \right(x)+e^x{f}^{\prime }(x\left)\right\}+c$

Q. .If π < x < 2π, prove that √1 + cos x + √1 − cos x / √1 + cos x – √1 − cos x = cot ( π /4 + x /2 )

Q. Which of the following integral represents the summation ${\sum }^{}\left(4x^3\right)\delta x$ for the limit $\delta x\to 0$?

1. 
   $\int \left(4x\right)dx$
2. 
   $\int \left(4x^2\right)$
3. 
   $\int \left(4x^3\right)dx$
4. 
   $\int \left(4x^4\right)dx$

Q. $\int \frac{e^x}{2e^x-5e^{-x}}dx$ is equal to

$\int \frac{e^x}{2e^x-5e^{-x}}dx$ का मान है

1. $\ln \left|2e^{2x}-5\right|+c$
2. $\ln \left|2e^x-5e^{-x}\right|+c$
3. $\frac{1}{4}\ln \left|2e^x-5e^{-x}\right|+c$
4. $\frac{1}{4}\ln \left|2e^{2x}-5\right|+c$

Q. what is derivative of cos (sin x)

Q. Evaluate:

1. Tan pi/8
2. Sin17pi/8

Q. The anti – derivative of the function (3x + 4) |sin x|, when $0<x<\pi$, is given by

1. $3sinx-(3x+4)cosx+c$
2. $3sinx+(3x+4)cosx+c$
3. $-3sinx-(3x+4)cosx+c$
4. None of these

Q.

differentiate x2 +sinx divided by xcosx