# Evaluating the definite integral ${\int_{-4}^{4}} (10x^{9} + 7x^{5}) dx$

$${\int_{-4}^{4}} (10x^{9} + 7x^{5}) dx$$

I got 2097152 as the answer, but the website I'm doing my homework on says it is wrong. Just need a little help here.

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You performed your computation incorrectly. In any case, there is a simpler way... – copper.hat Feb 2 '13 at 6:17

Observe the function is odd (meaning $f(-x)=-f(x)$), so try splitting the integral $$\int_{-4}^4 f(x)dx=\int_{-4}^0 f(x)dx+\int_0^4f(x)dx=\int_0^4f(-x)dx+\int_0^4f(x)dx.$$ What happens next?

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Although the odd function property is a better way: Just to add to the possible solutions: \begin{align} \int_{-4}^410x^9+7x^5&=\left[x^{10}+ \dfrac{7x^6}{6} \right] _{-4}^4\\ &=\left[4^{10}+ \dfrac{7\times 4^6}{6} \right]-\left[(-4)^{10}+ \dfrac{7(-4)^6}{6} \right]\\ \end{align}

\begin{align} &=\left[4^{10}+ \dfrac{7\times 4^6}{6} \right]-\left[(4)^{10}+ \dfrac{7(4)^6}{6} \right] \\&=0\end{align}

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Note that the function is an odd function and you integrating it from $-4$ to $4$. In general, if $f(x)$ is odd and integrable, we have $$\int_{-a}^a f(x) dx = 0$$where $a \in \mathbb{R}$.

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Note that you are solving $\int_{-a}^a f(x) dx$ where $f$ is odd. What does that say about $\int_{-a}^0 f(x) dx$ and $\int_{0}^a f(x) dx$?

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I messed up. I totally forgot about odd function. thank you! – Ak47 Feb 2 '13 at 6:39
I just did the integration and noticed at the end. Hindsight is 20-20. – copper.hat Feb 2 '13 at 6:43