1)INT ((x^2)\(xsinx +cosx)^2)dx ???
2)INT (sinx)\(1+sinx)dx????
3)INT 1\(cosx(1+cosx))dx ?????
4)INT x^2(1+lnx)dx ??????
5)INT (arcsine^x)\e^x ???????
thanks a lot.....
Some of these I cannot solve, but I'll try those that I can:
2) INT sinx/(1+sinx) dx
Multiply the numerator and denominator by 1 - sinx
INT [sinx(1 - sinx)]/(1 - sin^2x) dx
INT (sinx - (1 - cos^2x))/cos^2x dx
INT sinx/cos^2x dx - INT 1/cos^2x dx + INT cos^2x/cos^2x dx
INT secx*tanx dx - INT sec^2x dx + INT 1 dx
secx - tanx + x + C
4) INT x^2(1 + lnx) dx
Let u = 1 + lnx <--> du = 1/x dx
Let dv = x^2 dx <--> v = 1/3*x^3
(1 + lnx)(1/3*x^3) - 1/3 INT (x^3)(1/x) dx
1/3x^3(1 + lnx) - 1/3(1/3)x^3 + C
1/3x^3(1 + lnx) - 1/9x^3 + C
I'll try some of the others in a little bit
Wow! That was an interesting way to do it ... not one I would have thought of. Our answers seem completely different. Are they? (I would check, but I'm not sure how I could come up with the function you have from the one I have).
What inspired you to use those particular substitutions?
3) INT 1/(cosx(1 + cosx)) dx
Add and subtract cosx from the numerator:
INT (1 + cosx - cosx)/(cosx(1 + cosx)) dx
Separate the fraction
INT (1 + cosx)/(cosx(1 + cosx)) dx - INT cosx/(cosx(1 + cosx)) dx
INT 1/cosx dx - INT 1/(1 + cosx) dx
INT secx dx - INT 1/(1 + cosx) dx
ln|secx + tanx| - INT 1/(1 + cosx) dx
From this, multiply the numerator and denominator of the integration by 1 - cosx:
INT (1 - cosx)/(1 - cos^2x) dx
INT (1 - cosx)/sin^2x dx
INT 1/sin^2x dx - INT cosx/sin^2x dx
INT csc^2x dx - INT cscx*cotx dx
-cotx + cscx
ln|secx + tanx| + cotx - cscx + C
1) INT x^2/(x*sinx + cosx)^2 dx
Check to make sure this one is written correctly. It may be solvable as is, but it would be far easier if it were:
INT x^2/(sinx + cosx)^2 dx ... without the x being multiplied to the sinx.
I'm not going to attempt this problem any further until you can confirm that it is written correctly.
I took the derivative of this equation and it doesn't seem to work. I ended up asking a professor at my school to help me with this problem. He gave me the following:
INT x^2/(xsinx + cosx)^2 dx
Multply the numerator and denominator by cosx:
INT (x^2*cosx)/[cosx(xsinx + cosx)^2) dx
Now, use integration by parts:
Let u = x/cosx <--> du = (cosx + xsinx)/cos^2x dx
Let dv = (xcosx)/(xsinx + cosx)^2 dx <--> v = INT xcosx/(xsinx + cosx)^2 dx = -1/(xsinx + cosx) ... (to obtain this, see next step)
To find v, we need to do a substitution in the above integration:
Let n = xsinx + cosx <--> dn = (xcosx + sinx - sinx) dx --> dx = 1/(xcosx) dn
INT 1/n^2 dn = -1/n = -1/(xsinx + cosx)
From the integration by parts, we get:
-x/[cosx(xsinx + cosx)] + INT [(xsinx + cosx)/cos^2x](1/(xsinx + cosx)) dx
-x/[cosx(xsinx + cosx)] + INT 1/cos^2x dx
-x/[cosx(xsinx + cosx)] + tanx + C