The integral $\int x^n\,dx$ (for $n\neq -1$) equals
A$n\,x^{n-1}+C$
B$x^{n+1}+C$
C$\dfrac{x^{n-1}}{n-1}+C$
D$\dfrac{x^{n+1}}{n+1}+C$
Answer & Solution
Correct answer: D. $\dfrac{x^{n+1}}{n+1}+C$
1. The power rule for integration: increase the exponent by 1, divide by the new exponent.
2. $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$, valid for $n\neq -1$.
3. Check: $\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right)=\dfrac{(n+1)x^n}{n+1}=x^n$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.289_
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