Mathematics

Evaluate the given integral.
$$\displaystyle \int { \cfrac { { e }^{ x }\left( 1+x \right)  }{ \cos ^{ 2 }{ \left( x{ e }^{ x } \right)  }  }  } dx$$


ANSWER

$$\tan { \left( x{ e }^{ x } \right) } +C$$


SOLUTION
$$I=\displaystyle\int{\dfrac{{e}^{x}\left(1+x\right)}{{\cos}^{2}{\left(x{e}^{x}\right)}}dx}$$

Let $$t=x{e}^{x}\Rightarrow\,dt=x{e}^{x}+{e}^{x}\,dx=\left(x+1\right){e}^{x}dx={e}^{x}\left(1+x\right)dx$$

$$\Rightarrow\,I=\displaystyle\int{\dfrac{dt}{{\cos}^{2}{t}}}$$

$$\Rightarrow\,I=\displaystyle\int{{\sec}^{2}{t}\,dt}$$

Let $$u=\tan{t}\Rightarrow\,du={\sec}^{2}{t}\,dt$$

$$\Rightarrow\,I=\displaystyle\int{du}$$

$$\Rightarrow\,I=u+c$$ 

$$\Rightarrow\,I=\tan{t}+c$$     ............where $$u=\tan{t}$$

$$\Rightarrow\,I=\tan{\left(x{e}^{x}\right)}+c$$   ...........where $$t=x{e}^{x}$$

View Full Answer

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
$$\int\limits_{ - \pi /2}^{\pi /2} {\sin \left| x \right|} \,dx = $$
  • A.
  • B. 2
  • C. 4
  • D. 1

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
$$ \displaystyle \int { \frac { dx }{ x{ \left( { x }^{ 2 }+1 \right)  }^{ 2 } }  } =$$
  • A. $$ \displaystyle \ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } -\frac { 3 }{ 2\left( { x }^{ 2 }+1 \right) } +K$$
  • B. $$ \displaystyle -\ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 3 }{ 2\left( { x }^{ 2 }+1 \right) } +K$$
  • C. $$ \displaystyle -\ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 3 }{ 2\left( { x }+1 \right) } +K$$
  • D. $$ \displaystyle \ln { \frac { \left| x \right| }{ \sqrt { { x }^{ 2 }+1 } } } +\frac { 1 }{ 2\left( { x }^{ 2 }+1 \right) } +K$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate : $$\displaystyle\int \left(\dfrac{1}{\log x} - \dfrac{1}{(\log x)^2} \right) . dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate:- $$\int \dfrac{e^{2x} - 1}{e^{2x} + 1}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$n \space\epsilon \space N$$ & the A.M., G.M., H.M. & the root mean square of $$n$$ numbers $$2n+1, 2n+2, ...,$$ up to $$n^{th}$$ number are $$A_{n}$$, $$G_{n}$$, $$H_{n}$$ and $$R_{n}$$ respectively. 
On the basis of above information answer the following questions

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer