Check the solution to "Solve. ∫78x25−12x+4x2"

banganX

Answered question

2020-12-15

Solve.

\int \frac{78 x}{25 - 12 x + 4 x^{2}}

SoosteethicU

Skilled2020-12-16Added 102 answers

The given integration is

\int \frac{78 x}{25 - 12 x + 4 x^{2}}

A simple calculation the integration becomes

\int \frac{78 x}{25 - 12 x + 4 x^{2}} = 7 \int \frac{d x}{25 - 12 x + 4 x^{2}}

= 7 \int \frac{d x}{4 x^{2} - 12 x + 25}

= 7 \int \frac{d x}{4 (x^{2} - 3 x + \frac{25}{4})}

= \frac{7}{4} \int \frac{d x}{x^{2} - 3 x + \frac{25}{4}}

= \frac{7}{4} \int \frac{d x}{x^{2} - 3 x + \frac{25}{4}}

= \frac{7}{4} \int \frac{d x}{x^{2} - 2 \cdot x \cdot \frac{3}{2} + \frac{9}{4} + \frac{25}{4} - \frac{9}{4}}

= \frac{7}{4} \int \frac{d x}{{(x - \frac{3}{2})}^{2} + \frac{16}{4}}

= \frac{7}{4} \int \frac{d x}{{(x - \frac{3}{2})}^{2} + {(2)}^{2}}

= \frac{7}{4} \times \frac{1}{2} \tan^{- 1} (\frac{x - \frac{3}{2}}{2}) + C

[By using the formula \int \frac{d x}{x^{2}} + a^{2} = \frac{1}{a} \tan^{- 1} (\frac{x}{a}) + C]

= \frac{7}{8} \tan^{- 1} (\frac{2 x - 3}{4}) + C

Where C is the constant of integration.
Thus, the value of the integration is

\int \frac{7 d x}{25 - 12 x + 4 x^{2}} = \frac{7}{8} \tan^{- 1} (\frac{2 x - 3}{4}) + C

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