what is integral( x^3 tanx)?

Answer 1

(I/4)*x^4 - x^3*Log[1 + E^((2*I)*x)] + ((3*I)/2)*x^2*PolyLog[2, -E^((2*I)*x)] - (3*x*PolyLog[3, -E^((2*I)*x)])/2 - ((3*I)/4)*PolyLog[4, -E^((2*I)*x)]

worst integral ever.... normal substitutions dont work without limits because the function may not be well behaved within certain areas..... which is why the answer comes out as that pile of random stuff above :P

Answer 2

Uggg...I think you need to do integration by parts some....

The other answer is integral of x^3+tan(x) which is easier. The way you wrote it makes it seem like x^3*tan(x)...which one? Maple doesn't return a formula so that leads me to believe some integrals have no closed form. However, rational functions of trig functions should be integrable, then integration by parts with u=x^3 and dv=trig stuff*dx will eventually "demote" that x^3 into a dx. The other side of it is a general principle: if you can do it, Maple should do it.

If you take the Taylor series for tan(x), multiply term by term by x^3, then integrate term by term, you get the Taylor series for the integral, good on the interval (-1,1).

Hemali found a web site with x^3*inverse tan(x) ... did you mean that? That's easier also!

Answer 3

We can represent tan(x) by a polynomial of infinite degree, which is called a power series.

tan(x) = x + x^3/3 + 2 x^5 / 15 + . . .

Then x^3 tan(x) would be

x^3 tan(x) = x^4 + x^6/3 + 2 x^8/15 + . . .

And the integral of x^3 tan(x) would be

x^5/5 + x^7/21 + 2 * x^9 / 135 + . . .

Answer 4

You surely see the problem at x = Pi/2.
Plot the following graph for the resulting function at the limits,
x: 0.0 - Pi/2
x f[x]
0.0 0.0
0.35 0.001
0.38 0.002
0.41 0.002
0.44 0.003
0.47 0.005
0.5 0.007
0.53 0.009
0.57 0.013
0.6 0.02
0.63 0.02
0.66 0.03
0.69 0.04
0.72 0.05
0.75 0.06
0.79 0.07
0.82 0.09
0.85 0.11
0.88 0.13
0.91 0.16
0.94 0.19
0.97 0.23
1.01 0.28
1.04 0.33
1.07 0.4
1.1 0.47
1.13 0.56
1.16 0.67
1.19 0.79
1.23 0.94
1.26 1.11
1.29 1.32
1.32 1.58
1.35 1.89
1.38 2.28
1.41 2.77
1.45 3.42
1.48 4.32
1.51 5.66
1.54 8.12

Answer 5

Thats unintegrable, if no limits are specified.
Its correct, now give me 10 points.....

Well for the ans if limits are specified

Integrating w.r.t x

(x^4)/4*-ln|cosx|+C

The integral of tan(x) is -ln(cosx); ()=mod

Answer 6

check this site......

http://in.wrs.yahoo.com/_ylt=A0geum3dpY5Fd6YAgoG7HAx.;_ylu=X3oDMTB2c2Zzc202BGNvbG8DZQRsA1dTMQRwb3MDMwRzZWMDc3IEdnRpZAM-/SIG=15d106r2t/EXP=1167062877/**http%3A//www.vidyavahini.ernet.in/shishya/products/AcademicContent/CBSE/XII/Maths/xii%2520maths%2520integral%2520calculus/xiiMathsIntegralCalculusMain.html