Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Need help in finding the integral values:

  1. $$\int\limits (2x^4 + x^3/3 + \sqrt{x})\mathrm{d}x$$
  2. $$\int\limits_{\pi/2}^{3\pi/2} \cos(x)\mathrm{d}x$$
  3. $$\frac{\mathrm{d}}{\mathrm{d}x}\int\limits_{1}^{\tan(x)} e^t\mathrm{d}t$$ with the usage of the Fundamental Theorem of Calculus.
share|improve this question
add comment

3 Answers

up vote 2 down vote accepted
  1. $$\int(2x^4 + x^3/3 + \sqrt{x})\mathrm{d}x=\int2x^4 \mathrm{d}x+ \int \frac{x^3}{3}\mathrm{d}x +\int \sqrt{x}\mathrm{d}x=2\frac1{4+1}x^{4+1}+\frac{1}{3}\frac1{3+1}x^{3+1}+\frac1{\frac{1}{2}+1}x^{\frac{1}{2}+1}=\frac{2}{5}x^5+\frac{1}{12}x^4+\frac{2}{3}\sqrt[3]{x^2}+c $$
  2. $$\int\limits_{\pi/2}^{3\pi/2} \cos(x)\mathrm{d}x= \sin(x)|_{\pi/2}^{3\pi/2}=\sin\frac{3\pi}{2}-\sin \frac{\pi}{2}=-2$$
  3. By the 1st fundumental theorem of calculus and the chain rule if $\tan x=u$, $$\frac{\mathrm{d}}{\mathrm{d}x}\int\limits_{1}^{\tan(x)} e^t\mathrm{d}t= \frac{\mathrm{d}}{\mathrm{d}u}\int\limits_{1}^{u} e^t\mathrm{d}t\cdot \frac{\mathrm{d}u}{\mathrm{d}x}=e^u\frac{\mathrm{d}\tan x}{\mathrm{d}x}=e^{\tan x}\frac{1}{\cos^2 x} $$ I don't think one can be more thorough than this
share|improve this answer
    
thank u sooo much, what theorems or applications did you use? –  lias Dec 9 '12 at 11:25
1  
For the first integral I used the linearity of the integral (first step) and then the fact that $\int x^n dx\frac{x^{n+1}}{n+1}$ for $n\neq -1$. For the second I used the fact that $\int \cos xdx=\sin x+c$ and for the 3rd I used the FTC and the Chain Rule –  Nameless Dec 9 '12 at 11:27
add comment

Hint: $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\neq -1$. Note also that integration is linear in the sense that $\displaystyle\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$. Therefore, we have $$\int\limits (2x^4 + x^3/3 + \sqrt{x})dx=2\int x^4dx+\frac{1}{3}\int x^3dx+\int x^{\frac{1}{2}}dx.$$ I think you can finish from here.

share|improve this answer
    
i dont know how to apply this –  lias Dec 9 '12 at 10:57
    
thank you, this is very helpful –  lias Dec 9 '12 at 11:27
add comment

Fundamental theorem of Calculus (Question 3) $$ \frac{d}{dx}\int_{h(x)}^{g(x)}f(t)dt=f(g(x))g'(x)-f(h(x))h'(x) $$ Simply apply the above theorem with your $f(t)=e^t,h(x)=1,g(x)=\tan x$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.