Integral of xlnx
The integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C, where C is the integration constant. We can evaluate this integral using the method of integration by parts. We use the method of integration by parts to find the integral of the product of two functions. To find the integral of xlnx, we can consider xlnx as the product of two functions x and lnx. Mathematically, we can write the formula for the integral of xlnx as ∫xlnx dx = (x^{2}/2) lnx  x^{2}/4 + C. We know that integration is the reverse process of differentiation and using the formula of the integration by parts, we can find the integral of xlnx.
Further in this article, we will find the integral of xlnx and derive its formula. We will also find the integral of functions involving xlnx and solve a few examples using the integral of xlnx for a better understanding of the concept.
1.  What is the Integral of xlnx? 
2.  Integral of xlnx Formula 
3.  Integral of xlnx Proof 
4.  Integral of x ln x by √(x^{2}  1) 
5.  FAQs on Integral of xlnx 
What is the Integral of xlnx?
The integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C, where C is the constant of integration. We can calculate the integration of xlnx using integration by parts which is one of the most commonly used and important methods of integration. We can write the integral of xlnx with symbols as ∫xlnx dx = (x^{2}/2) lnx  x^{2}/4 + C, where ∫ is the symbol of integration, dx indicates that the integration of xlnx is with respect to the variable x and C is the integration constant. As we know that integration is the reverse process of differentiation, therefore we can say that the integral of xlnx is nothing but the antiderivative of xlnx. In the next section, let us go through its formula:
Integral of xlnx Formula
The formula for the integral of xlnx is given by, ∫xlnx dx = (x^{2}/2) lnx  x^{2}/4 + C with C as the constant of integration. This formula can be derived using the integration by parts method of integration. The image given below shows the formula for the integral of xlnx:
Integral of xlnx Proof
Now that we know that the integral of xlnx is equal to x^{2 }lnx/2  x^{2}/4 + C, we will prove it using the method of integration by parts formula. For the integral of f(x) g(x), the formula is given by, ∫f(x) g(x) dx = f(x) ∫g(x) dx  ∫[f'(x) ∫g(x) dx] dx. For the integral of xlnx we will choose the first function and second function using the sequence ILATE (I  Inverse, L  Logarithmic, A  Algebraic, T  Trigonometric, E  Exponential), therefore we have f(x) = lnx and g(x) = x. So, we will need the integral of x and derivative of lnx formulas. Therefore, we can find the integral of xlnx as
∫xlnx dx = ln(x) ∫x dx  ∫[(lnx)' ∫x dx] dx
= lnx × x^{2}/2  ∫[(1/x) × x^{2}/2] dx
= (x^{2}/2) lnx  (1/2) ∫x dx
= (x^{2}/2) lnx  (1/2)(x^{2}/2) + C
= (x^{2}/2) lnx  x^{2}/4 + C
Hence, we have derived the formula for the integral of xlnx using the method of integration by parts.
Integral of x ln x by √(x^{2}  1)
To find the integral of x ln x by √(x^{2}  1), we will use the formula of the integration by parts as above. The formula for the integration by parts is given by, ∫f(x) g(x) dx = f(x) ∫g(x) dx  ∫[f'(x) ∫g(x) dx] dx. We assume f(x) = ln x and g(x) = x/√(x^{2}  1). Before evaluating the integral of x lnx / √(x^{2}  1), let us calculate the integral of x/√(x^{2}  1). To find this, assume u^{2} = (x^{2}  1). Now, differentiating both sides, we have 2u du = 2x dx which implies xdx = udu. So, we have
∫[x/√(x^{2}  1)] dx = ∫(u/u) du
= ∫du
= u + K
= √(x^{2}  1) + K  (1)
Next, we will calculate the integral of √(x^{2}  1)/x. To find this, assume v = √(x^{2}  1), then dv/dx = x/√(x^{2}  1) which implies dx = √(x^{2}  1) dv/x = v/√(v^{2} + 1) dv and √(x^{2}  1)/x = v/√(v^{2} + 1). Therefore, we have
∫[√(x^{2}  1)/x] dx = ∫ [v/√(v^{2} + 1)] [v/√(v^{2} + 1) dv]
= ∫[v^{2}/(v^{2} + 1)] dv
= ∫[(v^{2} + 1  1)/(v^{2} + 1)] dv
= ∫(v^{2} + 1)/(v^{2} + 1) dv  ∫1/(v^{2} + 1) dv
= ∫dv  ∫1/(v^{2} + 1) dv
= v  arctan(v) + M, where M is the integration constant.
= √(x^{2}  1)  arctan(√(x^{2}  1))  (2)
Now, we will calculate the integral of xlnx as
∫x lnx/√(x^{2}  1) dx = ∫[x/√(x^{2}  1)] lnx dx
= lnx ∫[x/√(x^{2}  1)] dx  ∫[(lnx)' ∫[x/√(x^{2}  1)] dx] dx
= ln x × (√(x^{2}  1))  ∫[(1/x) √(x^{2}  1)] dx  [From (1)]
= √(x^{2}  1) ln x  [√(x^{2}  1)  arctan(√(x^{2}  1))]  [From (2)]
= √(x^{2}  1) ln x  √(x^{2}  1) + arctan(√(x^{2}  1)) + C, where C is the integration of constant.
Hence, the integral of x lnx / √(x^{2}  1) is equal to √(x^{2}  1) ln x  √(x^{2}  1) + arctan(√(x^{2}  1)) + C.
Important Notes on Integral of xlnx
 The integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C.
 We can find the integration of xlnx using the integration by parts method.
☛ Related Topics:
Integral of xlnx Examples

Example 1: Find the definite integral of xlnx from 0 to 1.
Solution: We know that the formula for the integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C. We will put the limits 0 to 1 in the formula to find its definite integral.
\(\begin{align} \int_{0}^{1} x\ln x \ dx&= \left [ \frac{x^2}{2} \ln x  \frac{x^2}{4} + C \right ]_0^1\\&=\left ( \frac{1^2}{2} \ln 1  \frac{1^2}{4} + C \right )\left ( \frac{0^2}{2} \ln 0  \frac{0^2}{4} + C \right ) \\&=(0\frac{1}{4}+C)(00+C)\\&=\frac{1}{4}\end{align}\)
Answer: The definite integral of xlnx from 0 to 1 is equal to 1/4.

Example 2: What is the integral of x lnx + 1?
Solution: To find the integral of xlnx + 1, we will use the formula for the integral of the sum of functions given by, ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx. For the integral of xlnx + 1, take f(x) = xlnx and g(x) = 1. Therefore, we have
∫(xlnx + 1) dx = ∫xlnx dx + ∫dx
= (x^{2}/2) lnx  x^{2}/4 + x + C  [Because the integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C and integral of 1 is x + K]
Answer: ∫(xlnx + 1) dx = (x^{2}/2) lnx  x^{2}/4 + x + C
FAQs on Integral of xlnx
What is Integral of xlnx in Calculus?
The integral of xlnx is equal to (x^{2}/2) lnx  x^{2}/4 + C, where C is the integration constant. We can evaluate this integral using the method of integration by parts.
What is the Formula for the Integral of xlnx?
The formula for the integral of xlnx is given by, ∫xlnx dx = (x^{2}/2) lnx  x^{2}/4 + C with C as the constant of integration.
What is the Definite Integral of xlnx From 0 to 1?
The definite integral of xlnx from 0 to 1 is equal to 1/4. We can evaluate this integral by substituting the limits 0 and 1 into the formula of the integral of xlnx given by, (x^{2}/2) lnx  x^{2}/4 + C.
How to Find the Integral of xlnx?
We can calculate the integral of xlnx using the method of integration by parts. We can assume f(x) = lnx and g(x) = x and substitute into the formula ∫f(x) g(x) dx = f(x) ∫g(x) dx  ∫[f'(x) ∫g(x) dx] dx to find the integral of xlnx.
How to Calculate the Integral of xlnx  1?
We can calculate the integral of xlnx  1 using the formula ∫[f(x)  g(x)] dx = ∫f(x) dx  ∫g(x) dx. We can write the integral as ∫(xlnx  1) dx = ∫xlnx dx  ∫dx = (x^{2}/2) lnx  x^{2}/4  x + C.
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