What is the integral from 1 to 4 of 1/(2sqrt(x)+3) ?

The integral from 1 to 4 of 1/(2sqrt(x)+3) is (-3ln(7)+3ln(5))/2+1

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integral from 1 to 4 of 1/(2sqrt(x)+3)

\int_{1}^{4} \frac{1}{2 x + 3} d x

\frac{- 3 ln ( 7 ) + 3 ln ( 5 )}{2} + 1

Decimal

0.49529 \dots

Solution steps

\int_{1}^{4} \frac{1}{2 x + 3} d x

Apply u-substitution

= \int_{1}^{2} \frac{2 u}{2 u + 3} d u

Take the constant out:

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

= 2 \cdot \int_{1}^{2} \frac{u}{2 u + 3} d u

Apply u-substitution

= 2 \cdot \int_{5}^{7} \frac{v - 3}{4 v} d v

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int_{5}^{7} \frac{v - 3}{v} d v

Expand

\frac{v - 3}{v} : 1 - \frac{3}{v}

= 2 \cdot \frac{1}{4} \cdot \int_{5}^{7} 1 - \frac{3}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot \frac{1}{4} (\int_{5}^{7} 1 d v - \int_{5}^{7} \frac{3}{v} d v)

\int_{5}^{7} 1 d v = 2

\int_{5}^{7} \frac{3}{v} d v = 3 (ln (7) - ln (5))

= 2 \cdot \frac{1}{4} (2 - 3 (ln (7) - ln (5)))

Simplify

2 \cdot \frac{1}{4} (2 - 3 (ln (7) - ln (5))) : \frac{- 3 ln ( 7 ) + 3 ln ( 5 )}{2} + 1

= \frac{- 3 ln ( 7 ) + 3 ln ( 5 )}{2} + 1

What is the integral from 1 to 4 of 1/(2sqrt(x)+3) ?
The integral from 1 to 4 of 1/(2sqrt(x)+3) is (-3ln(7)+3ln(5))/2+1