Question: "Determine the following indefinite integral. ∫(4x−4x)dx"

Baublysiz

Answered question

2021-11-21

Determine the following indefinite integral.

\int (4 \sqrt{x} - \frac{4}{\sqrt{x}}) d x

Befoodly

Beginner2021-11-22Added 17 answers

Step 1: Given that
Determine the following indefinite integral.

\int (4 \sqrt{x} - \frac{4}{\sqrt{x}}) d x

Step 2: Solve

\int (4 \sqrt{x} - \frac{4}{\sqrt{x}}) d x = 4 \int \sqrt{x} - 4 \int \frac{1}{\sqrt{x}} d x

= 4 [\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}] - 4 [\frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}]

= \frac{8}{3} [x^{\frac{3}{2}}] - 8 \sqrt{x}

Step 3: Check by differentiation

y = \frac{8}{3} x^{\frac{3}{2}} - 8 x^{\frac{1}{2}}

\frac{d y}{d x} = \frac{8}{3} \times \frac{3}{2} \times x^{\frac{3}{2} - 1} - 8 \times \frac{1}{2} \times x^{\frac{1}{2} - 1}

\frac{d y}{d x} = 4 x^{\frac{1}{2}} - 4 x^{- \frac{1}{2}}

\frac{d y}{d x} = 4 \sqrt{x} - \frac{4}{\sqrt{x}}

Hence Verified.

inenge3y

Beginner2021-11-23Added 20 answers

Step 1: Expand.

\int 4 \sqrt{x} - \frac{4}{\sqrt{x}} d x

Step 2: Use Sum Rule:

\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x

\int 4 \sqrt{x} d x - \int \frac{4}{\sqrt{x}} d x

Step 3: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

\frac{8 x^{\frac{3}{2}}}{3} - \int \frac{4}{\sqrt{x}} d x

Step 4: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\frac{8 x^{\frac{3}{2}}}{3} - 4 \int \frac{1}{\sqrt{x}} d x

Step 5: Since

\frac{1}{\sqrt{x}} = x^{- \frac{1}{2}}

, using the Power Rule,

\int x^{- \frac{1}{2}} d x = 2 x^{\frac{1}{2}}

\frac{8 x^{\frac{3}{2}}}{3} - 8 \sqrt{x} + C

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