# How do you evaluate the integral intx^3+4x^2+5 dx?

Aug 24, 2014

Because this equation only consists of terms added together, you can integrate them separately and add the results, giving us:

$\int {x}^{3} + 4 {x}^{2} + 5 \mathrm{dx} = \int {x}^{3} \mathrm{dx} + \int 4 {x}^{2} \mathrm{dx} + \int 5 \mathrm{dx}$

Each of these terms can be integrated using the Power Rule for integration, which is:

$\int {x}^{n} \mathrm{dx} = {x}^{n + 1} / \left(n + 1\right) + C$

Plugging our 3 terms into this formula, we have:

$\int {x}^{3} \mathrm{dx} = {x}^{3 + 1} / \left(3 + 1\right) = {x}^{4} / 4$

$\int 4 {x}^{2} \mathrm{dx} = \frac{4 {x}^{2 + 1}}{2 + 1} = \frac{4 {x}^{3}}{3}$

$\int 5 \mathrm{dx} = \int 5 {x}^{0} \mathrm{dx} = \frac{5 {x}^{0 + 1}}{0 + 1} = \frac{5 {x}^{1}}{1} = 5 x$

Now we arrive at our final answer by adding these together, remembering to add our constant ($C$) on the end:

$\int {x}^{3} + 4 {x}^{2} + 5 \mathrm{dx} = {x}^{4} / 4 + \frac{4 {x}^{3}}{3} + 5 x + C$