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Okay I like have no idea how to do this. I tried integration by parts then everything came back uglier. I then put this in wolfram it says "No found in standard mathematical functions"

Is there a easy way to do this?

Maybe like FTOC II?

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    Where is this monster coming from ? – 2017-02-13
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    calculus 2 term test from last year (had an average of 41%). I did every single integral question from assignments / textbook that were assigned + more, and I only got 4/7 of the integration questions on here. – 2017-02-13
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    https://gyazo.com/5e603dd1bd15ebb279befcd9517d461d The exact picture. Above it just says Evaluate. – 2017-02-13
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    Ok, stupid me, I missed the obvious: odd function. – 2017-02-13

3 Answers 3

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For any odd function,

$$\int_{x=-a}^af(x)\,dx=\int_{y=a}^{-a}f(-y)\,(-dy)=-\int_{y=-a}^{a}f(y)\,dy,$$

then $I=-I=0$.

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    Ohhh so from what I'm understanding, if f(x) is odd then $f(-2) == f(2)$, and this would use the integral property $$\int_{a}^{a} f(x) dx = 0$$. Right? – 2017-02-13
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    @user349557: no, $f(-2)=-f(2)$. What you may say is $F(x)$ even, then $F(-2)=F(2)$. And no, the bounds do not become equal. – 2017-02-13
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Hint:

$I=\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$

$\implies I+I=\int_a^b[f(x)+f(a+b-x)]dx$

Here $a=-2,b=2$

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    So uhm I've never done definite integral integration that had me put the upper limit and lower limit in the function before. Could you explain further? – 2017-02-13
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Remember that the integral of an odd function from $−A$ to $+A$ is zero. Now, you have

$$f(x)=\sin(x^5)e^{x^8\sin(x^4)}$$

$$f(-x)=\sin(-x^5)e^{x^8\sin(x^4)}=-\sin(x^5)e^{x^8\sin(x^4)}=-f(x)$$

As the function is odd, the integral is zero.