Evaluate the integral \$ \int (2x^3 + 4x^2 - 3x + 5) dx\$.

The power rule

\$\int x^n = x^(n+1)/(n+1) + c \$

Splitting the integral, then use the power rule of integration

\$ \int 2x^3 dx + \int 4x^2 dx - \int 3x dx + \int 5 dx\$
⇒ \$ (2x^(3+1))/(3+1) + (4x^(2+1))/(2+1) - (3x^(1+1))/(1+1) + 5x + C \$

After simplification

⇒ \$ (cancel(2)^1x^4)/(\cancel4^2) + 4/3 x^3 - 3/2 x^2 + 5x + C \$
⇒ \$ (1/2)x^4 + (4/3)x^3 - (3/2)x^2 + 5x + "C" \$

The integral is \$ (1/2)x^4 + (4/3)x^3 - (3/2)x^2 + 5x + "C" \$.

This option shows the indefinite integral of the given function, obtained by integrating each term of \$(2x^3, 4x^2, -3x, and 5)\$ using the power rule

\$ (1/2)x^4 + (4/3)x^3 - (3/2)x^2 + 5x + C \$

This option is incomplete. It integrated only some terms in the original expression and is missing coefficients for integrated terms of -3x and 5

\$ 6x^2 + 8x - 3x + C \$

Option similar to answer (A) but with sign errors in some terms. Integrated terms of \$x^3\$ and \$x^2\$ have flipped signs compared to the correct solution

\$ (1/2)x^4 - (4/3)x^3 - (2/3)x^2 + 5x + C \$

Incorrect coefficients for \$x^4\$ and \$x^3\$, signs are correct, but constants are double what they should be in the correct solution

\$ 2x^4 + 4x^3 - 3x^2 + 5x + C \$

Option

A

Can you summarize what you’ve understood in the above steps?