# Homework Help: Stuck with integration

1. Apr 11, 2005

### UrbanXrisis

$$\int_a^b f(x)dx = a+2b$$
$$\int_a^b (f(x)+5)dx =?$$
$$\int_a^b (f(x)+5)dx =\int_a^b f(x)dx+\int_a^b 5dx$$
$$a+2b+\int_a^b 5dx$$

I'm stuck, what should be my next step?

2. Apr 11, 2005

### HallsofIvy

Are you serious? You can't integrate $$\int_a^b5dx$$?

Isn't that the same as $$5\int_a^b dx= 5(b-a)$$?

Isn't that about the first thing you learned in integration?

3. Apr 11, 2005

### UrbanXrisis

hehe, thanks

what's the rule for $$\int e^x$$?
is it $$e^x \int x$$?

4. Apr 11, 2005

### dextercioby

Why?It's the exponential.Integration just adds a constant...Don't forget the differential of "x".

Daniel.

5. Apr 11, 2005

### UrbanXrisis

$$\int e^x$$ = $$\int e^x dx$$ ?

6. Apr 11, 2005

### dextercioby

No,no,the first notation is incorrect.It should always be

$$\int \ \mbox{function} \cdot \mbox{element of integration}$$

Daniel.

7. Apr 11, 2005

### UrbanXrisis

so if a question was $$\int e^{\frac{x}{2}} dx = e^{\frac{x}{2}} \int \frac{x}{2} dt$$

8. Apr 11, 2005

### HallsofIvy

$$\int e^x dx= e^x+ C$$

because $$\frac{d e^x}{dx}= e^x$$, of course.

9. Apr 11, 2005

### dextercioby

No,if course not.U need to make a substitution

$$\frac{x}{2}=u$$

Daniel.

EDIT:BTW,knowledge of integration techniques assumed knowledge of differentiation methodes.

10. Apr 11, 2005

### UrbanXrisis

$$\int e^{\frac{x}{2}} dx$$
$$u=x/2$$
$$du=1/2dx$$
$$\int e^{u} 2du$$
$$=2e^{\frac{x}{2}}$$

11. Apr 11, 2005

### HallsofIvy

Let u= x/2. Then 2u= x so 2du= dx.
$$\int e^{\frac{x}{2}}dx$$ becomes $$2\int e^u du= 2e^u+ C= 2e^{\frac{x}{2}}+ C$$

Surely you've learned simple substitutions.

$$e^x\int \frac{x}{2}dt$$, on the other hand, is $$e^x(\frac{x^2}{4}+ C)$$.

Last edited by a moderator: Apr 11, 2005
12. Apr 11, 2005

### dextercioby

That's right up to a constant,which should never be forgotten when computing indefinite integrals.

Daniel.

13. Apr 11, 2005

### UrbanXrisis

is $$cos^2(x)=cos(x)cos(x)$$
so..
$$\frac{d}{dx}cos^2(x)=-2sin(x)cos(x)$$

14. Apr 11, 2005

### dextercioby

Yeah,why?U could apply the chain rule as well.

Daniel.

15. Apr 11, 2005

### HallsofIvy

Yes, by golly!

(I think we are posting a cross purposes now!)

16. Apr 11, 2005

### UrbanXrisis

so if e is raised to any exponet that is not x, then I must use a subsitution when integrating?

17. Apr 11, 2005

### dextercioby

Yes.Always.Make that depends from case to case.Usually the antiderivatives of exponentials of "weird" arguments are not expressible in terms of "elementary functions".Simples example

$$\int e^{-x^{2}} \ dx$$

Daniel.

Last edited: Apr 11, 2005
18. Apr 12, 2005

### Jameson

Just out of curiousity, when I did that particular integral on Mathematica's Online Integrator, I got:

$$\int e^{-x^{2}} \ dx = \frac{1}{2}\sqrt{\pi} ERF[x] + C$$

Can someone tell me what Erf is?

19. Apr 12, 2005

### dextercioby

Sure

$$\mbox{erf} \ (x)=:\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} \ dt$$

Daniel.

20. Apr 12, 2005

### Jameson

Well, then that makes perfect sense. Is that something that is standardly used in Calculus? The constant seems to be random.