allery$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:
A$x^n \log x + C$
B$\dfrac{x^{n-1}}{n-1} + C$
C$\dfrac{x^{n+1}}{n+1} + C$
D$n x^{n-1} + C$
Answer & Solution
Correct answer: C. $\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = x^n$ for $n \ne -1$
so by anti-differentiation $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$.
The exception $n = -1$ is handled separately as $\int \dfrac{1}{x}\,dx = \log|x| + C$.
Keep practising Unit 6A: Indefinite Integrals & Fundamental Theorem
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