Question

(verification) still typing ....
Integral x+x^2tan(x) dx
It is a definite integral from "-pi/2" to "pi/2", but I am just checking if I can find the general antiderivative of this function.

Answer 1

it looks odd

Answer 2

\[\int\limits_{ }^{ } x+x^2\tan(x)~dx\]\[\frac{1}{2}x^2 + \int\limits_{ }^{ } x^2\tan(x)~dx\]\[\frac{1}{2}x^2 - x^2 \ln(\cos x) + \int\limits_{ }^{ } 2x \ln(\cos x)~dx\]so till now I am posting it.

Answer 3

when @ganeshie8 said it looks odd
he meant not weird :p

Answer 4

\[\int\limits_{ }^{ } \ln(\cos x ) dx\]\[x \ln(\cos x ) dx + \int\limits_{ }^{ }x \sin(x)\sec(x) dx\]\[x \ln(\cos x ) dx + \int\limits_{ }^{ }x \tan(x) dx\]\[x \ln(\cos x ) - x \ln(\cos x ) +\int\limits_{ }^{ } \ln(\cos x) dx\]so we have,
\[\int\limits_{ }^{ } \ln(\cos x ) dx =x \ln(\cos x) -x \ln(\cos x) + \int\limits_{ }^{ } \ln(\cos x ) dx \]

Answer 5

so the integral of ln(cosx) is zero? but that can't be.

Answer 6

you are doing a lot of work for nothing :p

Answer 7

I just want to find the antiderivative, I know the rule for an odd function.

Answer 8

\[\text{ if }f(-x)=-f(x) \text{ then } F(-x)=F(x) \text{ where } F'=f \\ \int\limits_{-a}^{a}f(x) dx=F(x)|_{-a}^{a}=F(a)-F(-a)=F(a)-F(a)=0\]

Answer 9

I probably did a mistake ln(cosx) dx in my second reply.

Answer 10

reply 2, line 2 and 3, it is a minus, not a plus.

Answer 11

I know knwo the odd function property. but I wanna solve it.

Answer 12

So I'll start again from\[\frac{1}{2}x^2 - x^2\ln(\cos x) + \int\limits_{ }^{ } 2x \ln (\cos x ) dx \]

Answer 13

http://www.wolframalpha.com/input/?i=integrate%28x%5E2*tan%28x%29%2Cx%29
it doesn't look like an elementary integral

Answer 14

actually, sorry, it is \[\int\limits_{ }^{ }x+x^2\tan(x)~dx~=\frac{1}{2}x^2 - x^2\ln(\cos x) + \int\limits_{ }^{ } 2x \ln (\cos x ) dx\]

Answer 15

not an elemetary one, but I am just challenging myself. But perhaps you are right, and I'll just give up.

Answer 16

Okay, I'll make it tan(x^3) lol.

Answer 17

\[\int\limits_{ }^{ }x+x^2\tan(x^3) dx\]\[\frac{1}{2}x+\frac{1}{3}\int\limits_{ }^{ }\tan(u)~du\]\[\frac{1}{2}x+\frac{1}{3}\ln \left| x^3 \right|+C\]\[\frac{1}{2}x+\ln \left| x \right|+C\]

Answer 18

but this is not fair though.

Answer 19

nvm