What is the integral from 0 to 1 of 2/((5+3x)^{1/2)} ?

The integral from 0 to 1 of 2/((5+3x)^{1/2)} is (2(4*2^{1/2}-2*5^{1/2}))/3

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integral from 0 to 1 of 2/((5+3x)^{1/2)}

\int_{0}^{1} \frac{2}{( 5 + 3 x ) ^{\frac{1}{2}}} d x

\frac{2 ( 4 \cdot 2 ^{\frac{1}{2}} - 2 \cdot 5 ^{\frac{1}{2}} )}{3}

Decimal

0.78981 \dots

Solution steps

\int_{0}^{1} \frac{2}{( 5 + 3 x ) ^{\frac{1}{2}}} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{1} \frac{1}{( 5 + 3 x ) ^{\frac{1}{2}}} d x

Apply Integration By Parts

: [\frac{x}{( 5 + 3 x ) ^{\frac{1}{2}}} - \int - \frac{3 x}{2 ( 5 + 3 x ) ^{\frac{3}{2}}} d x]_{0}^{1}

= 2 [\frac{x}{( 5 + 3 x ) ^{\frac{1}{2}}} - \int - \frac{3 x}{2 ( 5 + 3 x ) ^{\frac{3}{2}}} d x]_{0}^{1}

\int - \frac{3 x}{2 ( 5 + 3 x ) ^{\frac{3}{2}}} d x = - \frac{3 x + 10}{3 ( 3 x + 5 ) ^{\frac{1}{2}}}

= 2 [\frac{x}{( 5 + 3 x ) ^{\frac{1}{2}}} - (- \frac{3 x + 10}{3 ( 3 x + 5 ) ^{\frac{1}{2}}})]_{0}^{1}

Simplify

2 [\frac{x}{( 5 + 3 x ) ^{\frac{1}{2}}} - (- \frac{3 x + 10}{3 ( 3 x + 5 ) ^{\frac{1}{2}}})]_{0}^{1} : 2 [\frac{2}{3} (3 x + 5)^{\frac{1}{2}}]_{0}^{1}

= 2 [\frac{2}{3} (3 x + 5)^{\frac{1}{2}}]_{0}^{1}

Compute the boundaries:

\frac{4 \cdot 2 ^{\frac{1}{2}} - 2 \cdot 5 ^{\frac{1}{2}}}{3}

= 2 \cdot \frac{4 \cdot 2 ^{\frac{1}{2}} - 2 \cdot 5 ^{\frac{1}{2}}}{3}

Simplify

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

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What is the integral from 0 to 1 of 2/((5+3x)^{1/2)} ?
The integral from 0 to 1 of 2/((5+3x)^{1/2)} is (2(4*2^{1/2}-2*5^{1/2}))/3