xn,dx= xn+1n+1+C, n!= -1 includes rational and negative powers on real, defined intervals. Differentiating xn+1/(n+1) multiplies by n+1 and lowers the power; this reverses the rule. Rewrite surds and reciprocals as powers. Add one to the exponent, divide by the new exponent, and combine the terms with one constant. (sqrt(x)+x-2),dx= 23x3/2-x-1+C (x>0) Exam check: x-1 is excluded from the power formula; even roots require a non-negative radicand, odd real roots also allow negative inputs, and denominators cannot vanish.

\(\int x^n\,\mathrm{d}x=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1\) includes rational and negative powers on real, defined intervals.

Differentiating \(\dfrac{x^{n+1}}{n+1}\) multiplies by \(n+1\) and lowers the power; this reverses the rule.

Rewrite surds and reciprocals as powers. Add one to the exponent, divide by the new exponent, and combine the terms with one constant.

\(\int(\sqrt{x}+x^{-2})\,\mathrm{d}x=\frac23x^{\frac{3}{2}}-x^{-1}+C\quad(x>0)\)

Exam check: \(x^{-1}\) is excluded from the power formula; even roots require a non-negative radicand, odd real roots also allow negative inputs, and denominators cannot vanish.

Surds as powers

Surds as powers