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 two-third 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) x^3/2 + 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.

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) x^3/2 + 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) x^3/2 + C, where C is the constant of integration.

Now that we know that the integration of root x is equal to (2/3) x^3/2 + 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 = x^1/2. Therefore, we have

∫√x dx = ∫x^1/2 dx

= x^{1/2 + 1}/(1/2 + 1) + C

= x^3/2/(3/2) + C

= (2/3) x^3/2 + C

Hence, the integral of square root x is equal to (2/3) x^3/2 + C, where C is the integration constant.

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) x^3/2 + C. In this, we will substitute the limits to find its definite integral.

\(\begin{align} \int_{1}^{10}\sqrt{x} \ dx&= \left [ \frac{2}{3}x^{\frac{3}{2}} \right ]_1^{10}\\&=\frac{2}{3}(10)^{\frac{3}{2}} - \frac{2}{3}(1)^{\frac{3}{2}}\\&=\frac{2}{3}[10^{\frac{3}{2}}-1]\end{align}\)

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) x^3/2 + 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

☛ Related Topics:

• Integration of Sin 4x
• Integration of Sec 3x
• Integration of Tan Square x

What is Integration of Root x?

The integration of root x is (2/3) x^3/2 + 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 = x^1/2.

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)(10^3/2 - 1).