integral of (8x+1-9e^x) — step-by-step solution

Let's compute the integral:

\int(8 x + 1 - 9 e^{x})dx

We attempt to reverse an integration: starting from

- 9 e^{x}

and

{(\exp (a))}^{'} = \exp (a)

, we differentiate using exp

Let's perform the differentiation:

{(\exp (x))}^{'}

\exp (x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

- 9 e^{x} + \int(8 x + 1)dx

- 9 e^{x} + (\int(8 x)dx + \int(1)dx)

ℹ

\int(8 x)dx

8 \cdot \frac{x^{2}}{2}

‖

\int(1)dx

x

- 9 e^{x} + 8 \cdot \frac{x^{2}}{2} + x

ℹ

8 \cdot \frac{x^{2}}{2}

\frac{8 x^{2}}{2}

‖

\frac{8 x^{2}}{2}

4 x^{2}

- 9 e^{x} + 4 x^{2} + x

▶Indefinite integrals are defined up to an additive constant C, so this yields:

- 9 e^{x}

+ 4 x^{2}

+ x

+ C