How to evaluate $\int_{-10}^{10}(8x^9+5x^5)\,dx$ using the fundamental theorem of calculus?

Question

Can you somebody help me to solve this question?

Evaluate the intergral using the fundamental theorem of calculus.
$$\int_{-10}^{10}(8x^9+5x^5)\,dx$$

Answer 1

$\int_{-10}^{10}(8x^9+5x^5)dx=\int_{-10}^{10}8x^9dx+\int_{-10}^{10}5x^5dx=8\int_{-10}^{10}x^9dx+5\int_{-10}^{10}x^5)dx=(8\cdot\frac{x^{10}}{10}+5\cdot\frac{x^{6}}{6})|_{-10}^{10}=\frac{8}{10}(10^{10}-(-10)^{10})+\frac{5}{6}(10^6-(-10)^6)=\frac{8}{10}(10^{10}-10^{10})+\frac{5}{6}(10^6-10^6)=0$

Answer 2

(I will take this rather deliberately, you can certainly do it much faster.)

By the fundamental theorem of calculus, we have:

$$ \int_a^b f(x) \mathrm dx = F(b) - F(a) $$

Where: $$ F(x) = \int f(x) \mathrm dx $$

So:

$$\int 8x^9 + 5x^5 \mathrm dx$$

Break it into two parts:

$$ \int 8x^9 \mathrm dx + \int 5x^5 \mathrm dx $$

Move the constant outside of the integral:

$$ 8\int x^9 \mathrm dx + 5\int x^5 \mathrm dx $$

The power rule for integration gives: $$ 8 \cdot \frac{x^{10}}{10} + C_1 + 5 \cdot \frac{x^6}{6} + C_2 $$ $$ \frac{8x^{10}}{10} + \frac{5x^6}{6} + C_1 + C_2 $$

Merge the constants of integration: $$ \frac{8x^{10}}{10} + \frac{5x^6}{6} + C $$

So:

$$ \int_{-10}^{10} f(x) \mathrm dx = \frac{8 \cdot 10^{10}}{10} + \frac{5 \cdot 10^6}{6} + C - \left(\frac{8 \cdot (-10)^{10}}{10} + \frac{5 \cdot (-10)^6}{6} + C\right) $$

Note that the substitution $q = \frac{8 \cdot 10^{10}}{10} + \frac{5 \cdot 10^6}{6} + C$ makes it:

$$\int_{-10}^{10} f(x) \mathrm dx = q - q = 0$$

This holds for any odd function $g(x)$. Given:

$$g(-x) = -g(x)$$

We can show that if $G(x) = \int g(x) \mathrm dx$ (and $G(x)$ exists), then it is an even function. So:

$$\int_{-a}^a g(x) \mathrm dx = G(a) - G(-a) $$ Since $G(x)$ is odd, $G(a) = G(-a)$, so the integral is equal to $0$.

Answer 3

There is no reason to go through the process of integration. You can easily see that the expression $y=8x^9 +5x^5$ is a polynomial with only odd-degree terms, and there is no constant term. This polynomial is symmetric about the origin. The integral is taken from -10 to +10, equidistant on either side of the origin. There will be equal areas above as below the x-axis and the total sum of area will be zero.