Evaluate -d x5 x-22 x-7where C is constant of integration

Let I=∫d x(5 x 2)(2 x+7) ⇒ I=110∫d x(x 25)(x+72) ⇒ I=110· nbsp;1039∫(x+72) (x 25)(x 25)(x+7 nbsp; 2) d x ⇒ I=139∫d xx 25 139∫d xx+72 ⇒ I=139ln|x 25x+7 n ...

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Standard XII

Mathematics

Standard Formulae - 1

Evaluate ∫d x...

Question

Evaluate $\int \frac{d x}{(5 x - 2) (2 x + 7)}$
(where $C$ is constant of integration)

\frac{1}{39} ln ∣ ∣ ∣ \frac{7 x + 2}{2 x - 7} ∣ ∣ ∣ + C

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\frac{1}{39} ln ∣ ∣ ∣ \frac{7 x - 2}{5 x - 7} ∣ ∣ ∣ + C

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\frac{1}{39} ln ∣ ∣ ∣ \frac{5 x - 2}{2 x + 7} ∣ ∣ ∣ + C

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\frac{1}{39} ln ∣ ∣ ∣ \frac{5 x + 2}{2 x - 7} ∣ ∣ ∣ + C

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Solution

The correct option is C $\frac{1}{39} ln ∣ ∣ ∣ \frac{5 x - 2}{2 x + 7} ∣ ∣ ∣ + C$
Let $I = \int \frac{d x}{(5 x - 2) (2 x + 7)}$
$\Rightarrow I = \frac{1}{10} \int \frac{d x}{(x - \frac{2}{5}) (x + \frac{7}{2})}$
$\Rightarrow I = \frac{1}{10} \cdot \frac{10}{39} \int \frac{(x + \frac{7}{2}) - (x - \frac{2}{5})}{(x - \frac{2}{5}) (x + \frac{7}{2})} d x$

$\Rightarrow I = \frac{1}{39} \int \frac{d x}{x - \frac{2}{5}} - \frac{1}{39} \int \frac{d x}{x + \frac{7}{2}}$

$\Rightarrow I = \frac{1}{39} ln ∣ ∣ ∣ ∣ ∣ ∣ \frac{x - \frac{2}{5}}{x + \frac{7}{2}} ∣ ∣ ∣ ∣ ∣ ∣ + C$

$∴ I = \frac{1}{39} ln ∣ ∣ ∣ \frac{5 x - 2}{2 x + 7} ∣ ∣ ∣ + C$

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Standard Formulae - 1

Standard XII Mathematics

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Evaluate ∫dx(5x−2)(2x+7) (where C is constant of integration)

Evaluate $\int \frac{d x}{(5 x - 2) (2 x + 7)}$
(where $C$ is constant of integration)