Mathematics

# $\int 2^{2^{2^{x}}} 2^{2^{x}} 2^{x}dx$.

##### SOLUTION

$\int { { 2 }^{ { 2 }^{ { 2 }^{ x } } } } { 2 }^{ { 2 }^{ x } }{ 2 }^{ x }\quad dx$

Let ${ 2 }^{ x }=t\\ \cfrac { dt }{ dx } ={ 2 }^{ x }\ln { 2 } \\ { 2 }^{ x }dx=\cfrac { dt }{ \ln { 2 } } \\ { 2 }^{ { 2 }^{ t } }{ 2 }^{ t }\cfrac { dt }{ \ln { 2 } }$

Let ${ 2 }^{ t }=m\\ \cfrac { dm }{ dt } ={ 2 }^{ t }\ln { 2 } \\ { 2 }^{ t }dt=\cfrac { \left( dm \right) }{ \ln { 2 } } \\ \int { \cfrac { { 2 }^{ m }dm }{ { \left( \ln { 2 } \right) }^{ 2 } } } =\cfrac { 2^m }{ { \left( \ln { 2 } \right) }^{ 3 } } =\cfrac { { 2 }^{ { 2 }^{ { 2 }^{ x } } } }{ { \left( \ln { 2 } \right) }^{ 3 } } +C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 One Word Medium
Evaluate:$\displaystyle \int \frac{dx}{\sqrt{15-8x^{2}}}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate the given integral.
$\displaystyle \int { \left( \cfrac { 2-x }{ { (1-x) }^{ 2 } } \right) } { e }^{ x }dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate $\displaystyle\int _{ 0 }^{ \cfrac { \pi }{ 2 } }{ \cfrac { \sin { x } }{ 1+\cos { x } +\sin { x } } } dx\quad \quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int\frac{\sin^{-1}x-\cos^{-1}x}{\sin^{-1}x+\cos^{-1}x}dx=$
• A. $\log[\sin^{-1}x+\cos^{-1}x]+c$
• B. $\dfrac{4}{\pi}[x\sin^{-1}x+\sqrt{1-x^{2}}]+c$
• C. $\dfrac{4}{\pi}[x\sin^{-1}x-\sqrt{1-x^{2}}]+C$
• D. $\displaystyle \frac{4}{\pi}[x\sin^{-1}x+\sqrt{1-x^{2}}]-x+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.