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Integration of Root x
Integration of root x is determined using the formula of the integration given by ∫xn dx = xn+1/(n + 1) + C. Since root x is a radical, therefore we substitute n = 1/2 in the formula to get the integration of root x. The integral of square root x is equal to twothird of x raised to the power of three by two plus the integration constant. Mathematically, the integration of root x is written as ∫√x dx = (2/3) x3/2 + C.
Further in this article, we will derive the integral of square root x using the integration formula, and the integration of root x square plus a square and the integral of root x square minus a square. We will also solve solve some examples for a better understanding.
1.  What is Integration of Root x? 
2.  Integral of Square Root x Proof 
3.  Definite Integration of Root x 
4.  Integration of Root x Square Plus a Square 
5.  FAQs on Integration of Root x 
What is Integration of Root x?
The integration of root x is nothing but the integral of square root x with respect to x which is given by ∫√x dx = (2/3) x3/2 + C, where C is the constant of integration, ∫ denote the symbol of the integration and dx indicates the integral of square root x with respect to x. We can compute the integration of root x using the formula of integration ∫xn dx = xn+1/(n + 1) + C. Here we have n = 1/2 as root x is a radical expression. Hence, the formula for the integration of root x is given by, ∫√x dx = (2/3) x3/2 + C, where C is the constant of integration.
Integral of Square Root x Proof
Now that we know that the integration of root x is equal to (2/3) x3/2 + C, we will now prove it using the formula of integration. We will use the formula ∫xn dx = xn+1/(n + 1) + C by substituting n = 1/2 in it as √x is nothing but x raised to the power of one by two, that is, √x = x1/2. Therefore, we have
∫√x dx = ∫x1/2 dx
= x1/2 + 1/(1/2 + 1) + C
= x3/2/(3/2) + C
= (2/3) x3/2 + C
Hence, the integral of square root x is equal to (2/3) x3/2 + C, where C is the integration constant.
Definite Integration of Root x
Next, we will find the definite integration of root x with limits from 1 to 10. We know that the integral of the square root x formula is ∫√x dx = (2/3) x3/2 + C. In this, we will substitute the limits to find its definite integral.
\(\begin{align} \int_{1}^{10}\sqrt{x} \ dx&= \left [ \frac{2}{3}x^{\frac{3}{2}} \right ]_1^{10}\\&=\frac{2}{3}(10)^{\frac{3}{2}} – \frac{2}{3}(1)^{\frac{3}{2}}\\&=\frac{2}{3}[10^{\frac{3}{2}}1]\end{align}\)
Integration of Root x Square Plus a Square
In this section, we will find the integral of the square root of x square plus a square, that is, √(x2 + a2). To find this integral, we will use the method of integration by parts and the formula ∫1/√(x2 + a2) dx = log x + √(x2 + a2) + C. The formula for integration by parts is ∫f(x) g(x) dx = f(x) ∫g(x) dx – ∫[d(f(x))/dx ∫ g(x) dx] dx. Here f(x) = √(x2 + a2) and g(x) = 1 as we can write √(x2 + a2) as √(x2 + a2).1. Hence, we have
∫√(x2 + a2) dx = ∫√(x2 + a2).1 dx
⇒ ∫√(x2 + a2) dx = √(x2 + a2) ∫dx – ∫[d(√(x2 + a2))/dx ∫dx] dx
⇒ ∫√(x2 + a2) dx = x√(x2 + a2) – ∫x2/√(x2 + a2) dx
⇒ ∫√(x2 + a2) dx = x√(x2 + a2) – ∫(a2 – a2 + x2)/√(x2 + a2) dx
⇒ ∫√(x2 + a2) dx = x√(x2 + a2) + a2 ∫1/√(x2 + a2) dx – ∫(x2 + a2)/√(x2 + a2) dx
⇒ ∫√(x2 + a2) dx = x√(x2 + a2) + a2 ∫1/√(x2 + a2) dx – ∫√(x2 + a2) dx
⇒ ∫√(x2 + a2) dx + ∫√(x2 + a2) dx = x√(x2 + a2) + a2 ∫1/√(x2 + a2) dx
⇒ 2 ∫√(x2 + a2) dx = x√(x2 + a2) + a2[log x + √(x2 + a2) + C]
⇒ ∫√(x2 + a2) dx = (x/2)√(x2 + a2) + (a2/2) [log x + √(x2 + a2) + C]
⇒ ∫√(x2 + a2) dx = (x/2)√(x2 + a2) + (a2/2) log x + √(x2 + a2) + K, where K = C(a2/2)
Hence, the integral of root x square plus a square is given by, ∫√(x2 + a2) dx = (x/2)√(x2 + a2) + (a2/2) log x + √(x2 + a2) + K, where K is the integration constant. Similarly, we can determine the internal of root x square minus a square.
Integration of Root x Square Minus a Square
As we derived the integration of root x square plus a square, now we will determine the integration of root x square minus a square, that is, √(x2 – a2). We will use the formula of integration ∫1/√(x2 + a2) dx = log x + √(x2 + a2) + C and integration by parts. Therefore, we have
∫√(x2 – a2) dx = ∫√(x2 – a2).1 dx
⇒ ∫√(x2 – a2) dx = √(x2 – a2) ∫dx – ∫[d(√(x2 – a2))/dx ∫dx] dx
⇒ ∫√(x2 – a2) dx = x√(x2 – a2) – ∫x2/√(x2 – a2) dx
⇒ ∫√(x2 – a2) dx = x√(x2 – a2) – ∫(a2 – a2 + x2)/√(x2 – a2) dx
⇒ ∫√(x2 – a2) dx = x√(x2 – a2) – a2 ∫1/√(x2 – a2) dx – ∫(x2 – a2)/√(x2 – a2) dx
⇒ ∫√(x2 – a2) dx = x√(x2 – a2) – a2 ∫1/√(x2 – a2) dx – ∫√(x2 – a2) dx
⇒ ∫√(x2 – a2) dx + ∫√(x2 – a2) dx = x√(x2 – a2) – a2 ∫1/√(x2 – a2) dx
⇒ 2 ∫√(x2 – a2) dx = x√(x2 – a2) – a2[log x + √(x2 – a2) + C]
⇒ ∫√(x2 – a2) dx = (x/2)√(x2 – a2) – (a2/2) [log x + √(x2 – a2) + C]
⇒ ∫√(x2 – a2) dx = (x/2)√(x2 – a2) – (a2/2) log x + √(x2 – a2) + K, where K = C(a2/2)
Hence, the integral of root x square minus a square is given by, ∫√(x2 – a2) dx = (x/2)√(x2 – a2) – (a2/2) log x + √(x2 – a2) + K, where K is the integration constant.
Important Notes on Integration of Root x
 The integration of root x is ∫√x dx = (2/3) x3/2 + C, where C is the integration constant.
 ∫√(x2 – a2) dx = (x/2)√(x2 – a2) – (a2/2) log x + √(x2 – a2) + K
 ∫√(x2 + a2) dx = (x/2)√(x2 + a2) + (a2/2) log x + √(x2 + a2) + K
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Integration of Root x Examples

Example 1: Evaluate the integration of root x plus one by root x, that is, √x + 1/√x.
Solution: To determine the integration of √x + 1/√x, we will use the formula of the integral of square root x. Also, we will use the formula ∫xn dx = xn+1/(n + 1) + C.
∫[√x + 1/√x] dx = ∫√x dx + ∫(1/√x) dx
= (2/3) x3/2 + 2√x + C
Answer: Hence, the integration of root x plus one by root x is (2/3) x3/2 + 2√x + C.

Example 2: Determine the integral of square root x minus one, that is, √(x – 1).
Solution: To find the integral of √(x – 1), we will use the substitution method.
Assume √(x – 1) = u ⇒ 1/2√(x – 1) dx = du ⇒ dx = 2u du
∫√(x – 1) dx = ∫u 2u du
= ∫2u2 du
= 2u3/3 + C
= (2/3) [√(x – 1)]3 + C
= (2/3) (x – 1)3/2 + C
Answer: The integral of square root x minus one is (2/3) (x – 1)3/2 + C.
Integration of Root x Questions
FAQs on Integration of Root x
What is Integration of Root x?
The integration of root x is (2/3) x3/2 + C, where C is the constant of integration. It can be determined by using the formula ∫xn dx = xn+1/(n + 1) + C.
How to Find Integral of Square Root x?
The integral of square root x can be found using the formula of integration ∫xn dx = xn+1/(n + 1) + C. In this formula, we can substitute n = 1/2 as root x can be written as √x = x1/2.
What is the Integration of Root x Square Plus a Square?
The Integration of Root x Square Plus a Square is given by ∫√(x2 + a2) dx = (x/2)√(x2 + a2) + (a2/2) log x + √(x2 + a2) + K which is calculated using the integration by parts method of integration.
How to Find the Integral of Square Root x Square Minus a Square?
To find the integral of square root x square minus a square, we can use the method of integration by parts. The formula of this integration is ∫√(x2 – a2) dx = (x/2)√(x2 – a2) – (a2/2) log x + √(x2 – a2) + K.
What is the Value of Definite Integration of Root x From 1 to 10?
The value of the definite integration of root x with limits from 1 to 10 is (2/3)(103/2 – 1).
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