Mathematics

Solve:

$$\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$$


SOLUTION

$$I=\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$$

Let $$x+2=t^{2}$$. then , $$dx=2t dt$$

Also $$x=0\Rightarrow t^{2}=2$$ 

$$\Rightarrow t=\sqrt{2}$$ 

$$x=2\Rightarrow t^{2}=4$$ 

$$\Rightarrow t=2$$

$$\therefore I=\displaystyle\int_{\sqrt{2}}^{2}(t^{2}-2)\sqrt{t^{2}}2t dt$$ 

$$=2\displaystyle\int_{\sqrt{2}}^{2}(t^{4}-2t^{2})dt$$ 

$$=2\left[\dfrac{t^{5}}{5}-\dfrac{2t^{3}}{3}\right]_{\sqrt{2}}^{2}$$

$$\Rightarrow I=2\left[\left(\dfrac{32}{5}-\dfrac{16}{3}\right)-\left(\dfrac{4\sqrt{2}}{5}-\dfrac{4\sqrt{2}}{3}\right)\right]$$ 

$$=2\left(\dfrac{16}{15}+\dfrac{8\sqrt{2}}{15}\right)$$ 

$$=\dfrac{32+16\sqrt{2}}{15}$$
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 Single Correct Medium
$$\displaystyle\int^{2\pi}_0e^{x/2}\cdot \sin\left(\dfrac{x}{2}+\dfrac{\pi}{4}\right)dx=?$$
  • A. $$1$$
  • B. $$2\sqrt{2}$$
  • C. None of these
  • D. $$0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 One Word Medium
Solve $$\displaystyle\int \dfrac{ { e }^{ 2x }-{ e }^{ -2x } }{ { e }^{ 2x }+{ e }^{ -2x } } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
The value of $$\displaystyle \int_{0}^{\pi /2} \dfrac {\sin 2t}{\sin^{4}t + \cos^{4}t} dt $$
  • A. $$\pi$$
  • B. $$\dfrac {\pi}{3}$$
  • C. $$\dfrac {\pi}{4}$$
  • D. $$\dfrac {\pi}{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
If $$\displaystyle \int_{1}^{2} e^{x^2} dx= a$$, then $$\displaystyle \int_{e}^{e^4}\sqrt{\ln x} \:dx$$ is equal to
  • A. $$2e^4-2e-a$$
  • B. $$2e^4-e-2a$$
  • C. $$e^4-e-a$$
  • D. $$2e^4-e-a$$

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