Find [latex]\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{n+1}}{n+1}+c\right)[/latex] and hence deduce that [latex]\int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c[/latex].

Question 13.2: Find d / dx (x^n+1 / n + 1 + c) and hence deduce that ∫ x^......

Engineering Mathematics A foundation for Electronic, Electrical, Communications and Systems Engineers [1040491]

Find $\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x^{n+1}}{n+1}+c\right)$ and hence deduce that $\int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c$ .

Step-by-Step

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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.
Function, y(x)	Derivative, y′	Function, y(x)	Derivative, y′
constant	0	$\cos ^{-1}(a x+b)$	$\frac{-a}{\sqrt{1-(a x+b)^2}}$
$x^n$	$n x^{n-1}$
$\mathrm{e}^x$	$\mathrm{e}^x$	$\tan ^{-1}(a x+b)$	$\frac{a}{1+(a x+b)^2}$
$\mathrm{e}^{-x}$	$-\mathrm{e}^{-x}$
$\mathrm{e}^{a x}$	$a \mathrm{e}^{a x}$	$\sinh (a x+b)$	$a \cosh (a x+b)$
		$\cosh (a x+b)$	$a \sinh (a x+b)$
$\ln x$	$\frac{1}{x}$	$\tanh (a x+b)$	$a \operatorname{sech}^2(a x+b)$
$\sin x$	$\cos x$	$\operatorname{cosech}(a x+b)$	$-a \operatorname{cosech}(a x+b) \times$
$\cos x$	$-\sin x$		$\operatorname{coth}(a x+b)$
$\sin (a x+b)$	$a \cos (a x+b)$	$\operatorname{sech}(a x+b)$	$-a \operatorname{sech}(a x+b) \times$
$\cos (a x+b)$	$-a \sin (a x+b)$		$\tanh (a x+b)$
$\tan (a x+b)$	$a \sec ^2(a x+b)$	$\operatorname{coth}(a x+b)$	$-a \operatorname{cosech}^2(a x+b)$
$\operatorname{cosec}(a x+b)$	$-a \operatorname{cosec}(a x+b) \cot (a x+b)$	$\sinh ^{-1}(a x+b)$	$\frac{a}{\sqrt{(a x+b)^2+1}}$
$\sec (a x+b)$	$a \sec (a x+b) \tan (a x+b)$		$\frac{a}{\sqrt{(a x+b)^2-1}}$
$\cot (a x+b)$	$-a \operatorname{cosec}^2(a x+b)$	$\cosh ^{-1}(a x+b)$	$\frac{a}{\sqrt{(a x+b)^2-1}}$
$\sin ^{-1}(a x+b)$	$\frac{a}{\sqrt{1-(a x+b)^2}}$	$\tanh ^{-1}(a x+b)$	$\frac{a}{1-(a x+b)^2}$