Math Doubts

Proof of Power Rule of Integration

Let $x$ represents a variable, $n$ and $k$ represent constants. An algebraic expression $x^{n+1}+k$ is formed by the literals. Now, let’s derive the power rule of integration mathematically in integral calculus.

Define the Derivative of Algebraic function

The algebraic expression is defined in terms of $x$. So, it should be differentiated with respect to $x$ and it can be expressed in mathematical form as follows.

$\dfrac{d}{dx}{\,\Big(x^{n+1}+k\Big)}$

Differentiate the Algebraic expression

The algebraic expression is expressed in $x$. Hence, differentiate the algebraic expression with respect to $x$ for evaluating its derivative.

$\implies$ $\dfrac{d}{dx}{\,\Big(x^{n+1}+k\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\,\Big(x^{n+1}\Big)}$ $+$ $\dfrac{d}{dx}{\,(k)}$

The derivative of the first term can be evaluated by the power rule of differentiation and the derivative of the second term is zero as per the constant rule of differentiation.

$\implies$ $\dfrac{d}{dx}{\,\Big(x^{n+1}+k\Big)}$ $\,=\,$ $(n+1) \times x^{n+1-1}+0$

$\implies$ $\require{cancel} \dfrac{d}{dx}{\,\Big(x^{n+1}+k\Big)}$ $\,=\,$ $(n+1) \times x^{n+\cancel{1}-\cancel{1}}$

$\implies$ $\dfrac{d}{dx}{\,\Big(x^{n+1}+k\Big)}$ $\,=\,$ $(n+1)x^n$

Proving the Power Rule by inverse operation

It is evaluated that the derivative of the expression $x^{n+1}+k$ is $(n+1)x^n$. According to the inverse operation, the primitive or an anti-derivative of expression $(n+1)x^n$ is equal to $x^{n+1}+k$. It can be written in mathematical form as follows.

$\displaystyle \int{(n+1)x^n\,}dx$ $\,=\,$ $x^{n+1}+k$

The constant factor $n+1$ can be separated from the integral operation by the constant multiple rule of integration.

$\implies$ $(n+1) \times \displaystyle \int{x^n\,}dx$ $\,=\,$ $x^{n+1}+k$

Now, let us simplify the mathematical equation.

$\implies$ $\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}+k}{n+1}$

$\implies$ $\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}$ $+$ $\dfrac{k}{n+1}$

In the right hand side expression of the equation, the first term is a function in $x$ but the second term is a constant. So, it can be simply denoted by a constant $c$, which is known as the constant of integration.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}+c$

Thus, the power rule of integration is derived mathematically in differential calculus.

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