Novalee S.

asked • 10/28/24

Evaluating INtergrals

Evaluate the integral

∫ with 10 as top limit and 0 on bottom with (3)/ the fourth root of x dx

Julius N.

To evaluate the integral \[ \int_0^{10} \frac{3}{\sqrt[4]{x}} \, dx, \] we can rewrite the integrand to simplify the expression. Step 1: Rewrite the Integrand The fourth root of \( x \) can be expressed as \( x^{1/4} \). So, we have: \[ \frac{3}{\sqrt[4]{x}} = 3 \cdot x^{-\frac{1}{4}}. \] Thus, the integral becomes: \[ \int_0^{10} 3 \cdot x^{-\frac{1}{4}} \, dx. \] We can factor out the constant 3: \[ = 3 \int_0^{10} x^{-\frac{1}{4}} \, dx. \] Step 2: Integrate \( x^{-\frac{1}{4}} \) To integrate \( x^{-\frac{1}{4}} \), we use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n \neq -1 \). Here, \( n = -\frac{1}{4} \), so \( n + 1 = -\frac{1}{4} + 1 = \frac{3}{4} \). Applying the power rule: \[ \int x^{-\frac{1}{4}} \, dx = \frac{x^{\frac{3}{4}}}{\frac{3}{4}} = \frac{4}{3} x^{\frac{3}{4}}. \] Step 3: Substitute and Evaluate the Definite Integral Now we substitute back into our integral: \[ 3 \int_0^{10} x^{-\frac{1}{4}} \, dx = 3 \cdot \frac{4}{3} x^{\frac{3}{4}} \Big|_0^{10}. \] The \( 3 \) and \( \frac{4}{3} \) cancel out, leaving: \[ = 4 x^{\frac{3}{4}} \Big|_0^{10}. \] Now we evaluate this at the upper and lower limits: \[ = 4 \left(10^{\frac{3}{4}}\right) - 4 \left(0^{\frac{3}{4}}\right). \] Since \( 0^{\frac{3}{4}} = 0 \), this simplifies to: \[ = 4 \cdot 10^{\frac{3}{4}}. \] Final Answer So, the evaluated integral is: \[ 4 \cdot 10^{\frac{3}{4}}. \]
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10/29/24

William W.

tutor
Julius N., I hope you can see that your method of writing solutions does not come across well in the text-only window provided for comments and is not useful unless the reader understands LaTex
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10/29/24

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