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Popular >

integral of sqrt((2+3x)/(x-3))

Solution

\int \frac{2 + 3 x}{x - 3} d x

Solution

\frac{1}{3} (3 x + 2 3 x - 9 + 11 ln \frac{3 x - 9}{11} + \frac{3 x + 2}{11}) + C

Solution steps

\int \frac{2 + 3 x}{x - 3} d x

Simplify

\frac{2 + 3 x}{x - 3} : \frac{2 + 3 x}{x - 3}

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

Apply u-substitution

: \int \frac{u}{3 u - 11} d u

= \int \frac{u}{3 u - 11} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{u}{u - 11} d u

Apply u-substitution

: \int 2 v^{2} + 11 d v

= \frac{1}{3} \cdot \int 2 v^{2} + 11 d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot 2 \cdot \int v^{2} + 11 d v

Apply Trigonometric Substitution

: \int 11 sec^{3} (w) d w

= \frac{1}{3} \cdot 2 \cdot \int 11 sec^{3} (w) d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot 2 \cdot 11 \cdot \int sec^{3} (w) d w

Apply Integral Reduction

= \frac{1}{3} \cdot 2 \cdot 11 (\frac{sec ^{2} ( w ) sin ( w )}{2} + \frac{1}{2} \cdot \int sec (w) d w)

\int sec (w) d w = ln ∣ tan (w) + sec (w) ∣

= \frac{1}{3} \cdot 2 \cdot 11 (\frac{sec ^{2} ( w ) sin ( w )}{2} + \frac{1}{2} ln ∣ tan (w) + sec (w) ∣)

Substitute back

= \frac{1}{3} \cdot 2 \cdot 11 \frac{sec ^{2} ( arctan ( \frac{1}{11} 2 + 3 x - 11 ) ) sin ( arctan ( \frac{1}{11} 2 + 3 x - 11 ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{11} 2 + 3 x - 11)) + sec (arctan (\frac{1}{11} 2 + 3 x - 11))

Simplify

\frac{1}{3} \cdot 2 \cdot 11 \frac{sec ^{2} ( arctan ( \frac{1}{11} 2 + 3 x - 11 ) ) sin ( arctan ( \frac{1}{11} 2 + 3 x - 11 ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{11} 2 + 3 x - 11)) + sec (arctan (\frac{1}{11} 2 + 3 x - 11)) : \frac{1}{3} (3 x + 2 3 x - 9 + 11 ln \frac{3 x - 9}{11} + \frac{3 x + 2}{11})

= \frac{1}{3} (3 x + 2 3 x - 9 + 11 ln \frac{3 x - 9}{11} + \frac{3 x + 2}{11})

Add a constant to the solution

= \frac{1}{3} (3 x + 2 3 x - 9 + 11 ln \frac{3 x - 9}{11} + \frac{3 x + 2}{11}) + C

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