# 43 - Calculus Questions Answers

Find

lim n→∞ ⁿ√(n!) / n

**Asked By: LUFFY**

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**Joshi sir comment**

1 . Iₙ = ∫ ( 1/ ( x² + a²)ⁿ )dx

How to do this by substituting x = a tan Α

2. Is this true

∫ f(x) d(kg(x)) = k ∫ f(x) d(g(x))

where k is a constant

**Asked By: LUFFY**

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**Solution by Joshi sir**

d/dx((1+x^2+x^4)/(1+x+x^2))=ax+b

then a=?, b=?

**Asked By: RAJIV**

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**Joshi sir comment**

1+x^{2}+x^{4} = (1+x+x^{2})(1-x+x^{2})

now solve

Find a formula for a function g(x) satisfying the following conditions

a) domain of g is (-∞ , ∞ ) b) range of g is [-2 , 8] c) g has a period π d) g(2) = 3

**Asked By: VAIBHAV GUPTA**

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**Joshi sir comment**

g(x) = 3-5sin(2x-4)

Let f(x) = x^{135} + x^{125} - x^{115 }+ x^{5} +1. If f(x) is divided by x^{3}-x then the remainder is some function of x say g(x). Find the value of g(10)

**Asked By: VAIBHAV GUPTA**

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**Joshi sir comment**

for getting reminder put x^{3}= x so

x^{135} + x^{125} - x^{115 }+ x^{5} +1 will give

x^{45} + x^{41}*x^{2} - x^{38}*x +x*x^{2} +1

x^{15} + x^{13}*x^{2}*x^{2} - x^{12}*x^{2}*x +x*x^{2} +1

x^{17 }+ x^{3} +1

x^{5}*x^{2} + x + 1

x^{7 }+ x + 1

x^{2}*x + x +1

x^{3} + x +1

x+x+1

2x+1

now put x = 10

How to inegrate ( In t) dt /(t-1)

**Asked By: PARTHASARATHY**

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**Joshi sir comment**

let lnt = x so dt/t = dx so dt = tdx = e^{x}dx

now ∫(ln t)dt/(t-1) = ∫xe^{x}dx/(e^{x}-1) = ∫x(e^{x}-1+1)dx/(e^{x}-1)

now solve

∫√tanx

**Asked By: BONEY HAVELIWALA**

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**Joshi sir comment**

let √tanx = t

so sec^{2}xdx/2√tanx = dt

so (1+t^{4})dx/2t = dt

so dx = 2tdt/(1+t^{4})

now solve

if ƒ:R−{0} -> R, 2ƒ(x) − 3ƒ(1⁄x) = x² then ƒ(3)= ?

**Asked By: BONEY HAVELIWALA**

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**Joshi sir comment**

2ƒ(x) − 3ƒ(1⁄x) = x² (1)

so 2ƒ(1/x) − 3ƒ(x) = 1/x² (2)

multiply (1) to 2 and (2) to 3, then add the two equations, you will get f(x) then calculate f(3)

Sir/Madam,

Suddenly a question struck on my mind: 0×∞= 1 or 0?? as 1⁄0=∞.

**Asked By: BONEY HAVELIWALA**

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**Joshi sir comment**

it will be 0, assume it by considering that ∞ is a big number

for ex. 0*(1111111111111111111111111) = 0

FIND DOMAIN OF f(x)=log4log3 log2x WHERE 4,3,2 ARE BASES ?

**Asked By: SIDDHANT**

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**Joshi sir comment**

f(x)= log_{4}log_{3}log_{2}x

so 0 < log_{3}log_{2}x < ∞

so 1 < log_{2}x < ∞

so 2 < x < ∞