SL 5.5—Introduction to integration

Use $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for each term.

$\int 6x^{2}dx=2x^3$, $\int -4xdx=-2x^2$, $\int 3dx=3x$.

Adding these gives $2x^3-2x^2+3x$.

Since this is an indefinite integral, add $+C$, giving $2x^3-2x^2+3x+C$.

Option B forgot to divide by the new power after raising it, keeping the original coefficients unchanged.

Option C correctly integrates the first two terms but forgets to integrate the constant term $3$ into $3x$.

Option D differentiates the original function instead of integrating it.

Only option A correctly applies the power rule to every term and includes $+C$.

$\int 5x^{4}dx=\frac{5x^5}{5}=x^5$.

$\int xdx=\frac{x^2}{2}$.

Sum the results: $x^5+\frac{x^2}{2}$.

Include $+C$ for the indefinite integral, giving $x^5+\frac{x^2}{2}+C$.

Option B differentiates $5x^4$ instead of integrating it, giving $20x^3$.

Option C forgets to divide the second term by the new power, keeping $x^2$ instead of $\frac{x^2}{2}$.

Option D incorrectly divides by $4$ (the original power) instead of $5$ (the new power).

Only option A applies the power rule correctly to both terms.