Mathematics

If $$\int {\frac{{4{e^x} + 6{e^{ - x}}}}{{9{e^x} - 4{e^{ - x}}}}dx}  = Ax + B{\log _e}\left( {9{e^{2x}} - 4} \right) + C$$ then $$A=...... ,B=...... ,C=......$$.


SOLUTION
$$\displaystyle\int{\dfrac{4{e}^{x}+6{e}^{-x}}{9{e}^{x}-4{e}^{-x}}dx}$$

$$=\displaystyle\int{\dfrac{4{e}^{2x}+6}{9{e}^{2x}-4}dx}$$

We can write $$Numerator=A\left(Denominator\right)+B\dfrac{d}{dx}\left(Denominator\right)$$

$$4{e}^{2x}+6=A\left(9{e}^{2x}-4\right)+B\dfrac{d}{dx}\left(9{e}^{2x}-4\right)$$

$$4{e}^{2x}+6=A\left(9{e}^{2x}-4\right)+B\left(18{e}^{2x}\right)$$

$$4{e}^{2x}+6=\left(9A+18B\right){e}^{2x}+\left(-4A\right)$$

Comparing the coefficients, we have

$$-4A=6$$ or $$A=-\dfrac{6}{4}=-\dfrac{3}{2}$$

$$9A+18B=4\Rightarrow\,18B=4-9A=4+9\times\dfrac{3}{2}=\dfrac{8+27}{2}=\dfrac{35}{2}$$

$$\therefore\,B=\dfrac{35}{36}$$

$$\displaystyle\int{\dfrac{4{e}^{2x}+6}{9{e}^{2x}-4}dx}$$

$$=\displaystyle\int{\dfrac{A\left(9{e}^{2x}-4\right)+B\left(18{e}^{2x}\right)}{9{e}^{2x}-4}dx}$$

$$=\displaystyle\int{\dfrac{A\left(9{e}^{2x}-4\right)dx}{9{e}^{2x}-4}}+\displaystyle\int{\dfrac{B\left(18{e}^{2x}\right)dx}{9{e}^{2x}-4}}$$

$$=-\dfrac{3}{2}\displaystyle\int{\dfrac{\left(9{e}^{2x}-4\right)dx}{9{e}^{2x}-4}}+\dfrac{35}{36}\displaystyle\int{\dfrac{\left(18{e}^{2x}\right)dx}{9{e}^{2x}-4}}$$

$$=-\dfrac{3}{2}\displaystyle\int{dx}+\dfrac{35}{36}\displaystyle\int{\dfrac{\left(18{e}^{2x}\right)dx}{9{e}^{2x}-4}}$$

$$=-\dfrac{3}{2}x+\dfrac{35}{36}\log_{e}{\left(9{e}^{2x}-4\right)}+c$$

Comparing with $$Ax+B\log_{e}{\left(9{e}^{2x}-4\right)}+C$$

we get $$A=-\dfrac{3}{2},\,B=\dfrac{35}{36},\,C=c$$
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 Subjective Hard
Evaluate, $$\int_1^2 {\left( {3{x^2} - 1} \right)dx} $$ as the limit of a sum

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate : $$\displaystyle \int_{0}^{\pi /2}\sin^3 x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
The value of $$\displaystyle \int _{0}^{1}\dfrac{x^{4}(1-x)^{4}}{1+x^{2}}\ dx$$ is
  • A. $$\dfrac{2}{105}$$
  • B. $$\dfrac{22}{7}-\pi$$
  • C. $$\dfrac{71}{15}-\dfrac{3\pi}{2}$$
  • D. $$0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the integral $$\displaystyle\int_{-3}^{3}|x+1|\ dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Consider two differentiable functions $$f(x), g(x)$$ satisfying $$\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$$ & $$\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$$. where $$\displaystyle f(x)>0    \forall  x \in  R$$

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

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer