Mathematics

Let the equation of a curve passing through the point $$\displaystyle \left ( 0, 1 \right )$$ be given by $$\displaystyle y=\int x^{2}.e^{x^{3}}dx.$$ If the equation of the curve is written in the form $$\displaystyle x=f\left ( y \right )$$ then $$\displaystyle f\left ( y \right )$$ is


ANSWER

$$\displaystyle \sqrt[3]{\log _{e}\left ( 3y-2 \right )}$$


SOLUTION

Substitute $$x^3=t$$

$$ \Rightarrow 3x^2dx=dt$$

The equation becomes $$ \displaystyle y=\frac { 1 }{ 3 } \int { { e }^{ t }dt } \\ \displaystyle =\frac { 1 }{ 3 } { e }^{ t }+c\\ \displaystyle =\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c$$

Given that the curve passes through $$(0,1)$$. Using this we get

$$c=\dfrac{2}{3}$$

So, equation becomes $$ \displaystyle 3y=e^{x^3}+2$$

$$ \Rightarrow e^{x^3}=3y-2$$

$$ \Rightarrow x^3 = \ln{(3y-2)}$$

$$ \displaystyle \Rightarrow x=\sqrt[3]{ \ln{(3y-2)}}$$

View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Single Correct Hard
let $$f(x)=(x+1)^{2}+\frac{1}{x}$$ then the value of $$\int_{-2}^{1}f(x)(-x)dx$$ 
  • A. is equal to $$-\frac{81}{10}$$
  • B. is equal to $$\frac{81}{10}$$
  • C. does not exist
  • D. is equal to $$\dfrac{-15}{4}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
The value of $$\int\dfrac{{x^{2}}}{{(x^{2}+a^{2})(x^{2}+b^{2})}}dx$$ is
  • A. $$\dfrac{1}{b^{2}-a^{2}}[b\tan^{-1}]-a\tan^{-1}\dfrac{x}{a}]+C$$
  • B. $$\dfrac{1}{b^{2}-a^{2}}[a\tan^{-1}]-b\tan^{-1}\dfrac{x}{a}]+C$$
  • C. $$\dfrac{1}{b^{2}-a^{2}}[b\tan^{-1}]+a\tan^{-1}\dfrac{x}{a}]+C$$
  • D. $$none\ of\ these$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate the following integral
$$\int { \cfrac { { e }^{ 2x } }{ { e }^{ 2x }-2 }  } dx\quad $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
$$I = \int \dfrac {1}{x(x^{6}+1)}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 TRUE/FALSE Medium
If $$f,g,h$$ be continuous functions on $$[0,a]$$ such that $$f(a-x)=-f(x),g(a-x)=g(x)$$ and $$3h(x)-4h(a-x)=5$$ then  $$\displaystyle \int_0^a f(x)g(x)h(x)dx=0$$
  • A. False
  • B. True

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer