# Determine the following indefinite integral. \int (4\sqrt{x}-\frac{4}{\sqrt{x}})dx

Determine the following indefinite integral.
$\int \left(4\sqrt{x}-\frac{4}{\sqrt{x}}\right)dx$
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Befoodly
Step 1: Given that
Determine the following indefinite integral.
$\int \left(4\sqrt{x}-\frac{4}{\sqrt{x}}\right)dx$
Step 2: Solve
$\int \left(4\sqrt{x}-\frac{4}{\sqrt{x}}\right)dx=4\int \sqrt{x}-4\int \frac{1}{\sqrt{x}}dx$
$=4\left[\frac{{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]-4\left[\frac{{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right]$
$=\frac{8}{3}\left[{x}^{\frac{3}{2}}\right]-8\sqrt{x}$
Step 3: Check by differentiation
$y=\frac{8}{3}{x}^{\frac{3}{2}}-8{x}^{\frac{1}{2}}$
$\frac{dy}{dx}=\frac{8}{3}×\frac{3}{2}×{x}^{\frac{3}{2}-1}-8×\frac{1}{2}×{x}^{\frac{1}{2}-1}$
$\frac{dy}{dx}=4{x}^{\frac{1}{2}}-4{x}^{-\frac{1}{2}}$
$\frac{dy}{dx}=4\sqrt{x}-\frac{4}{\sqrt{x}}$
Hence Verified.
###### Not exactly what you’re looking for?
inenge3y
Step 1: Expand.
$\int 4\sqrt{x}-\frac{4}{\sqrt{x}}dx$
Step 2: Use Sum Rule: $\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$.
$\int 4\sqrt{x}dx-\int \frac{4}{\sqrt{x}}dx$
Step 3: Use Power Rule: $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$.
$\frac{8{x}^{\frac{3}{2}}}{3}-\int \frac{4}{\sqrt{x}}dx$
Step 4: Use Constant Factor Rule: $\int cf\left(x\right)dx=c\int f\left(x\right)dx$.
$\frac{8{x}^{\frac{3}{2}}}{3}-4\int \frac{1}{\sqrt{x}}dx$
Step 5: Since $\frac{1}{\sqrt{x}}={x}^{-\frac{1}{2}}$, using the Power Rule, $\int {x}^{-\frac{1}{2}}dx=2{x}^{\frac{1}{2}}$
$\frac{8{x}^{\frac{3}{2}}}{3}-8\sqrt{x}+C$