{"id":8879,"date":"2024-01-23T19:38:17","date_gmt":"2024-01-23T14:08:17","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8879"},"modified":"2024-01-23T19:38:22","modified_gmt":"2024-01-23T14:08:22","slug":"integration-of-root-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-root-x\/","title":{"rendered":"integration of root x"},"content":{"rendered":"\n

The integral of the root x, can be found using power rule. The integral of $\\sqrt x $ with respect to (x) is:<\/p>\n\n\n\n

\\[
\\int \\sqrt x \\, dx =\\frac{2}{3} x^{3\/2} + C
\\]<\/p>\n\n\n\n

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant.<\/p>\n\n\n\n

Proof of integration of root x<\/h2>\n\n\n\n

To integrate the square root of $x $ (i.e., $\\sqrt{x}$), we express it as $x^{1\/2} $. The integral is then:<\/p>\n\n\n\n

$$ \\int \\sqrt{x} \\, dx = \\int x^{1\/2} \\, dx $$<\/p>\n\n\n\n

To integrate this, we use the power rule for integration:<\/p>\n\n\n\n

$$ \\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C $$<\/p>\n\n\n\n

where $C $ is the constant of integration. Applying this rule:<\/p>\n\n\n\n

$$ \\int x^{1\/2} \\, dx = \\frac{x^{(1\/2) + 1}}{(1\/2) + 1} + C = \\frac{x^{3\/2}}{3\/2} + C $$<\/p>\n\n\n\n

Simplifying the fraction:<\/p>\n\n\n\n

$$ = \\frac{2}{3} x^{3\/2} + C $$<\/p>\n\n\n\n

<\/p>\n\n\n\n

So, the integral of (\\sqrt{x}$ is:<\/p>\n\n\n\n

$$ \\frac{2}{3} x^{3\/2} + C $$<\/p>\n\n\n\n

Integration based on root x<\/h2>\n\n\n\n

$$ \\int \\sqrt{ax + b} \\, dx = \\frac{2}{3a} (ax + b)^{3\/2} + C $$<\/p>\n\n\n\n

Proof<\/strong><\/p>\n\n\n\n

Choose a substitution that simplifies the integrand. Let ( $u = ax + b$ ). Then, ( $du = a \\, dx$ ) or ( $dx = \\frac{du}{a}$ ).
Substitute ( u ) and ( dx ) into the integral:
$$= \\int \\sqrt{u} \\cdot \\frac{du}{a} $$
$$= \\frac{1}{a} \\int u^{1\/2} \\, du $$
$$= \\frac{1}{a} \\cdot \\frac{2}{3} u^{3\/2} + C $$
Here, ( C ) is the constant of integration.
Substitute back for ( u ) to get the final answer:
$$\\int \\sqrt{ax + b} \\, dx= \\frac{2}{3a} (ax + b)^{3\/2} + C $$<\/p>\n\n\n\n

Definite Integral of root x<\/h2>\n\n\n\n

To evaluate a definite integral of the root x, $\\int_a^b \\sqrt x \\, dx$, where $ a $ and $ b $ are the limits of integration, we follow a similar process as with the indefinite integral and the we’ll apply the limits at the end.<\/p>\n\n\n\n

$$
\\int_a^b \\sqrt(x) \\, dx = [ \\frac{2}{3} x^{3\/2}]_a^b
$$<\/p>\n\n\n\n

Solved Examples<\/h2>\n\n\n\n

Question 1<\/strong><\/p>\n\n\n\n

$$ \\int \\sqrt{3x – 1} \\, dx $$<\/p>\n\n\n\n

Solution<\/strong><\/p>\n\n\n\n

Choose a substitution that simplifies the integrand. Let $ u = 3x – 1 $. Then, $ du = 3 \\, dx $ or $ dx = \\frac{du}{3} $.
Substitute $ u $ and $ dx $ into the integral:
$$ =\\int \\sqrt{u} \\cdot \\frac{du}{3} $$
You can simplify this to:
$$ =\\frac{1}{3} \\int u^{1\/2} \\, du $$
Now integrate with respect to $ u $:
$$ =\\frac{1}{3} \\cdot \\frac{2}{3} u^{3\/2} + C $$
Here, $ C $ is the constant of integration.
Substitute back for $ u $ to get the final answer:
$$= \\frac{2}{9} (3x – 1)^{3\/2} + C $$<\/p>\n\n\n\n

Question 2<\/strong><\/p>\n\n\n\n

$$ \\int \\sqrt{x} \\cdot \\ln(x) \\, dx $$<\/p>\n\n\n\n

Solution<\/strong><\/p>\n\n\n\n

Integrating $ \\sqrt{x} \\cdot \\ln(x) $ requires the use of integration by parts.<\/p>\n\n\n\n

$$ \\int u \\, dv = uv – \\int v \\, du $$<\/p>\n\n\n\n