Question 13.2: Find d / dx (x^n+1 / n + 1 + c) and hence deduce that ∫ x^......
Find \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{n+1}}{n+1}+c\right) and hence deduce that \int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c.
From Table 10.1 we find
\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{n+1}}{n+1}+c\right)=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{n+1}}{n+1}\right)+\frac{\mathrm{d}}{\mathrm{d} x}(c) \quad \begin{aligned}& \text { using the linearity } \\& \text { of differentiation }\end{aligned}
=\frac{1}{n+1} \frac{\mathrm{d}}{\mathrm{d} x}\left(x^{n+1}\right)+\frac{\mathrm{d}}{\mathrm{d} x}(c) \quad \begin{aligned}& \text { again using the } \\& \text { linearity of differentiation }\end{aligned}
\begin{aligned}& =\frac{1}{n+1}\left\{(n+1) x^n\right\}+0 \quad \text { using Table } 10.1 \\& =x^n\end{aligned}
Consequently, reversing the process we find
\int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c
as required. Note that this result is invalid if n = −1 and so this result could not be applied to the integral \int(1 / x) d x.
| Table 10.1 Derivatives of commonly used functions. |
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| Function, y(x) | Derivative, y′ | Function, y(x) | Derivative, y′ |
| constant | 0 | \cos ^{-1}(a x+b) | \frac{-a}{\sqrt{1-(a x+b)^2}} |
| x^n | n x^{n-1} | ||
| \mathrm{e}^x | \mathrm{e}^x | \tan ^{-1}(a x+b) | \frac{a}{1+(a x+b)^2} |
| \mathrm{e}^{-x} | -\mathrm{e}^{-x} | ||
| \mathrm{e}^{a x} | a \mathrm{e}^{a x} | \sinh (a x+b) | a \cosh (a x+b) |
| \cosh (a x+b) | a \sinh (a x+b) | ||
| \ln x | \frac{1}{x} | \tanh (a x+b) | a \operatorname{sech}^2(a x+b) |
| \sin x | \cos x | \operatorname{cosech}(a x+b) | -a \operatorname{cosech}(a x+b) \times |
| \cos x | -\sin x | \operatorname{coth}(a x+b) | |
| \sin (a x+b) | a \cos (a x+b) | \operatorname{sech}(a x+b) | -a \operatorname{sech}(a x+b) \times |
| \cos (a x+b) | -a \sin (a x+b) | \tanh (a x+b) | |
| \tan (a x+b) | a \sec ^2(a x+b) | \operatorname{coth}(a x+b) | -a \operatorname{cosech}^2(a x+b) |
| \operatorname{cosec}(a x+b) | -a \operatorname{cosec}(a x+b) \cot (a x+b) | \sinh ^{-1}(a x+b) | \frac{a}{\sqrt{(a x+b)^2+1}} |
| \sec (a x+b) | a \sec (a x+b) \tan (a x+b) | \frac{a}{\sqrt{(a x+b)^2-1}} | |
| \cot (a x+b) | -a \operatorname{cosec}^2(a x+b) | \cosh ^{-1}(a x+b) | \frac{a}{\sqrt{(a x+b)^2-1}} |
| \sin ^{-1}(a x+b) | \frac{a}{\sqrt{1-(a x+b)^2}} | \tanh ^{-1}(a x+b) | \frac{a}{1-(a x+b)^2} |