@Azimuddin_Sheikh It's my pleasure....

# Thread to share Good Questions from Calculus which requires deep thinking

**Rohan_Shinde1**#329

I am feeling bored today so let's do something exciting. So buckle up guys for some very good questions in maths coming up right next to you. Let's start with an easy one

Q.1)Evaluate \int \frac {dx}{\prod_{r=0}^n (x+r)}

**Rohan_Shinde1**#330

Q.2) Evaluate \int_1^{\infty} \frac {\left\{x\right\} -\frac 12}{x} dx

(Some people might have seen this earlier.)

**Rohan_Shinde1**#333

Q.3) Evaluate in terms of a and b the value of

\int_0^{\frac {\pi}
{2}}\ln(a^2\cos^2\theta+b^2\sin^2\theta)d\theta

For all a, b\in R -\{0\}

**Rohan_Shinde1**#334

Q.4) Evaluate in terms of k the value of

\displaystyle \int_0^{\pi} \frac {\cos (kx)}{5-4\cos x}dx

**Rohan_Shinde1**#335

Q.5) For all integers n we define \phi_n as follows

\phi_n=1 if n\equiv 0\pmod 4 or n\equiv 1\pmod 4

and

\phi_n= -1 if n\equiv 2\pmod 4 or n\equiv 3\pmod 4

For all non negative integers n, let

\displaystyle f(n)=\phi_0\binom {n}{0} +\phi_1\binom {n}{1}+\phi_2\binom {n}{2}+..... +\phi_n\binom {n}{n}

Find \displaystyle \sum_{n=0}^{\infty} \frac {f(n)}{n!}

**Rohan_Shinde1**#338

Q.6) Evaluate

\displaystyle \int_0^1 \left(\left\{ \frac {1}{x}\right\}\{2x\}\right) \frac {dx}{x}

**Azimuddin_2019**#339

@Rohan_Shinde1 bro , the question which I posted can be solved in one sec by just substituting 1+xsinx = t we will get limits from 1 to 1 so area zero

**Rohan_Shinde1**#340

Q.7) Let P_1(z)=-\ln(1-z) and

\displaystyle P_{k+1}(z) =\int_0^z \frac {P_k(t)}{t} dt

Then find P_4(1)

**Azimuddin_2019**#341

@Rohan_Shinde1 bro can u post some nice algebra problems in another thread ( of jee advance level) ??

**Rohan_Shinde1**#342

It's good to some substitution but also try to make sure it's inverse exist and the inverse of y=1+x\sin x is differentiable at every point in the limits of integral.

About the algebra questions I will surely try to create such thread.

**Rohan_Shinde1**#344

Q.8) Evaluate

\displaystyle \int_{\frac 12}^1 \frac {\ln x}{x-1}dx

Enjoy the questions ladies and gentlemen

**Rohan_Shinde1**#346

So did you check whether the inverse exists and whether or not it is differentiable?