Resolved: "Find the indefinite integral and check the result..."

pedzenekO

Answered question

2021-11-07

Find the indefinite integral and check the result by differentiation.

\int (2 - \frac{3}{x^{10}}) d x

Alara Mccarthy

Skilled2021-11-08Added 85 answers

Step 1
Consider the provided integral,

\int (2 - \frac{3}{x^{10}}) d x

Find the indefinite integral and check the result by differentiation.
So,

\int (2 - \frac{3}{x^{10}}) d x = \int 2 d x - \int \frac{3}{x^{10}} d x

Step 2
Therefore,

\int 2 d x = 2 x + C_{1}

\int \frac{3}{x^{10}} d x = \frac{3 x^{- 10 + 1}}{- 10 + 1} + C_{2} \Rightarrow \frac{1}{- 3 x^{9}} + C_{2}

Let

C = C_{1} - C_{2}

. Now, find the indefinite integral.

\int (2 - \frac{3}{x^{10}}) d x = \int 2 d x - \int \frac{3}{x^{10}} d x

= 2 x + C_{1} + \frac{1}{3 x^{9}} - C_{2}

= 2 x + \frac{1}{3 x^{9}} + C

Step 3
Now, differentiate the above

2 x + \frac{1}{3 x^{9}} + C

.
So,

\frac{d}{d x} (2 x + \frac{1}{3 x^{9}} + C) = \frac{d}{d x} (2 x) + \frac{d}{d x} (\frac{1}{3 x^{9}}) + \frac{d}{d x} (C)

= 2 + \frac{- 9 x^{- 9 + 1}}{3} + 0

= 2 - 3 x^{- 10}

= 2 - \frac{3}{x^{10}}

Hence,

\int (2 - \frac{3}{x^{10}}) d x

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