Best solutions to - "Find the indefinite integral. ∫x2+2x−6x4dx"

Oberlaudacu

Answered question

2021-12-12

Find the indefinite integral.

\int \frac{x^{2} + 2 x - 6}{x^{4}} d x

reinosodairyshm

Beginner2021-12-13Added 36 answers

Step 1
Consider the integral:

i = \int \frac{(x^{2} + 2 x - 6)}{x^{4}} d x

...(1)
Step 2
Rewrite (1) as:

I = \int (\frac{x^{2}}{x^{4}} + \frac{2 x}{x^{4}} - \frac{6}{x^{4}}) d x

= \int (x^{2 - 4} + 2 x^{1 - 4} - 6 x^{- 4}) d x

= \int (x^{- 2} + 2 x^{- 3} - 6 x^{- 4}) d x

Step 3
Solve the above integral using formula

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

I = (\frac{x^{- 2 + 1}}{- 2 + 1} + \frac{2 x^{- 3 + 1}}{- 3 + 1} - \frac{6 x^{- 4 + 1}}{- 4 + 1}) + C

= (- x^{- 1} - \frac{2 x^{- 2}}{2} + \frac{6 x^{- 3}}{3}) + C

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

= \frac{(2 - x - x^{2})}{x^{3}} + C

where C is a constant.

Linda Birchfield

Beginner2021-12-14Added 39 answers

\frac{(x^{2} + 2 x - 6)}{x^{4}} d x

We represent the original integral as the sum of the table of integrals:

\int \frac{x^{2} + 2 x - 6}{x^{4}} d x = \int x^{- 2} d x + \int \frac{2}{x^{3}} d x + \int - \frac{6}{x^{4}} d x

\int x^{- 2} d x

This tabulated integral:

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

\int \frac{2}{x^{3}} d x

it is tabulated integral:

\int \frac{2}{x^{3}} d x = - \frac{1}{x^{2}} + C

\int (- \frac{6}{x^{4}}) d x

it is tabulated integral:

\int - \frac{6}{x^{4}} d x = \frac{2}{x^{3}} + C

\int \frac{x^{2} + 2 x - 6}{x^{4}} d x = - \frac{1}{x} - \frac{1}{x^{2}} + \frac{2}{x^{3}} + C

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