Solved example of tabular integration
We can solve the integral $\int x^4\sin\left(x\right)dx$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$
Find the derivative of $x^4$ with respect to $x$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a function multiplied by a constant ($12$) is equal to the constant times the derivative of the function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
The derivative of the constant function ($24$) is equal to zero
Derive $P(x)$ until it becomes $0$
Find the integral of $\sin\left(x\right)$ with respect to $x$
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
Any expression multiplied by $1$ is equal to itself
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
Integrate $T(x)$ as many times as we have had to derive $P(x)$, so we must integrate $\sin\left(x\right)$ a total of $5$ times
With the derivatives and integrals of both functions we build the following table
Then the solution is the sum of the products of the derivatives and the integrals according to the previous table. The first term consists of the product of the polynomial function by the first integral. The second term is the product of the first derivative by the second integral, and so on.
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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