Mathematics

Evaluate $$\displaystyle \int \dfrac{e^x(1+x)}{\cos^2(xe^x)}dx$$ on $$I \subset R$$ \ $$\{x\in R:\cos(xe^x)=0\} $$


SOLUTION
$$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx}$$
Let
$$x \; {e}^{x} = u$$
$$\left( {e}^{x} + x \; {e}^{x} \right) dx= du$$
$$\Rightarrow {e}^{x} \left( 1 + x \right) dx = du$$
Thus the integral will become-
$$\int{\cfrac{du}{\cos^{2}{u}}}$$
$$= \int{\sec^{2}{u} \; du}$$
$$= \tan{u} + C$$
Substituting the value of $$u$$ in above equation, we get
$$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx} = \tan{\left( x \; {e}^{x} \right)} + C$$
Hence
$$\int{ \cfrac{{e}^{x} \left( 1 + x \right)}{\cos^{2}{\left( x \; {e}^{x} \right)}} \; dx} = \tan{\left( x \; {e}^{x} \right)} + C$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 One Word Hard
Evaluate:
$$\displaystyle\int_{-2}^{0}{\left({x}^{3}+3{x}^{2}+3x+3+\left(x+1\right)\cos{\left(x+1\right)}\right)dx}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium

Evaluate the following definite integral:

$$\displaystyle\int_{1}^{2} \dfrac 2x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate $$\displaystyle \int_{0}^{2}(x^2-x)dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Solve: 
$$\int_{}^{} {\frac{{\cos x}}{{\sqrt {{{\sin }^2}x - 2\sin x - 3} }}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$$  &  $$\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$$

On the basis of above information, answer the following questions: 

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer