Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { { x }^{ 3 }-3{ x }^{ 2 }+5x-7+{ x }^{ 2 }{ a }^{ x } }{ 2{ x }^{ 2 } }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{{x}^{3}-3{x}^{2}+5x-7+{x}^{2}{a}^{x}}{2{x}^{2}}dx}$$

$$=\dfrac{1}{2}\displaystyle\int{\dfrac{{x}^{3}-3{x}^{2}+5x-7+{x}^{2}{a}^{x}}{{x}^{2}}dx}$$

$$=\dfrac{1}{2}\displaystyle\int{x\,dx}-\dfrac{3}{2}\displaystyle\int{dx}+\dfrac{5}{2}\displaystyle\int{\dfrac{dx}{x}}-\dfrac{7}{2}\displaystyle\int{\dfrac{dx}{{x}^{2}}}+\dfrac{1}{2}\displaystyle\int{{a}^{x}\,dx}$$

$$=\dfrac{{x}^{2}}{4}-\dfrac{3x}{2}+\dfrac{5}{2}\log{\left|x\right|}+\dfrac{7}{2x}+\dfrac{{a}^{x}}{2\log{a}}+c$$ where $$c$$ is the constant of integration.

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 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Hard
Evaluate the integral, $$\int _{ 0 }^{ 1 }{ \cos { \left( 2\cot ^{ -1 }{ \sqrt { \dfrac { 1-x }{ 1+x }  }  }  \right)  }  } dx=$$
  • A. $$1/2$$
  • B. $$0$$
  • C. $$1$$
  • D. $$-1/2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
If $$I_{n} = \int_{0}^{\pi/4}\tan^{n} xdx$$ then $$\displaystyle \lim_{n\rightarrow \infty} n(I_{n} + I_{n - 2}) =$$
  • A. $$1/2$$
  • B. $$\infty$$
  • C. $$0$$
  • D. $$1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
$$\displaystyle  \int _{ 0 }^{ 1 }{ \tan ^{ -1 }{ \left( \dfrac { x }{ \sqrt { 1-{ x }^{ 2 } }  }  \right)  } dx }$$
  • A. $$\dfrac{\pi}{2}$$
  • B. $$0$$
  • C. None of these
  • D. $$\dfrac{\pi}{2}-1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate: $$\int\limits_0^{\pi /2} {\log } \left( {\tan \;x} \right)\;dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
$$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer