What is the integral of 2x6^{1/2 x} ?

The integral of 2x6^{1/2 x} is 8((3^{x/2}*2^{1/2 (x-2)}x)/(ln(6))-1/(ln^2(6))*6^{x/2})+C

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integral of 2x6^{1/2 x}

\int 2 x 6^{\frac{1}{2} x} d x

8 (\frac{3 ^{\frac{x}{2}} \cdot 2 ^{\frac{1}{2} (x - 2)} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{\frac{x}{2}}) + C

Solution steps

\int 2 x \cdot 6^{\frac{1}{2} x} d x

Take the constant out:

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

= 2 \cdot \int x \cdot 6^{\frac{1}{2} x} d x

Simplify

\frac{1}{2} x : \frac{x}{2}

= 2 \cdot \int x \cdot 6^{\frac{x}{2}} d x

Apply u-substitution

: \int 3^{u} \cdot 2^{u + 2} u d u

= 2 \cdot \int 3^{u} \cdot 2^{u + 2} u d u

3^{u} 2^{u + 2} u = 2^{2} \cdot 3^{u} \cdot 2^{u} u

= 2 \cdot \int 2^{2} \cdot 3^{u} \cdot 2^{u} u d u

Take the constant out:

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

= 2 \cdot 2^{2} \cdot \int 3^{u} \cdot 2^{u} u d u

Simplify

= 2 \cdot 4 \cdot \int 3^{u} \cdot 2^{u} u d u

Simplify

3^{u} \cdot 2^{u} u : 6^{u} u

= 2 \cdot 4 \cdot \int 6^{u} u d u

Apply Integration By Parts

: \frac{6 ^{u} u}{ln ( 6 )} - \int \frac{6 ^{u}}{ln ( 6 )} d u

= 2 \cdot 4 (\frac{6 ^{u} u}{ln ( 6 )} - \int \frac{6 ^{u}}{ln ( 6 )} d u)

\int \frac{6 ^{u}}{ln ( 6 )} d u = \frac{6 ^{u}}{ln ^{2} ( 6 )}

= 2 \cdot 4 (\frac{6 ^{u} u}{ln ( 6 )} - \frac{6 ^{u}}{ln ^{2} ( 6 )})

Substitute back

u = \frac{x}{2}

= 2 \cdot 4 (\frac{6 ^{\frac{x}{2}} \cdot \frac{x}{2}}{ln ( 6 )} - \frac{6 ^{\frac{x}{2}}}{ln ^{2} ( 6 )})

Simplify

2 \cdot 4 (\frac{6 ^{\frac{x}{2}} \cdot \frac{x}{2}}{ln ( 6 )} - \frac{6 ^{\frac{x}{2}}}{ln ^{2} ( 6 )}) : 8 (\frac{3 ^{\frac{x}{2}} \cdot 2 ^{\frac{1}{2} (x - 2)} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{\frac{x}{2}})

= 8 (\frac{3 ^{\frac{x}{2}} \cdot 2 ^{\frac{1}{2} (x - 2)} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{\frac{x}{2}})

Add a constant to the solution

= 8 (\frac{3 ^{\frac{x}{2}} \cdot 2 ^{\frac{1}{2} (x - 2)} x}{ln ( 6 )} - \frac{1}{ln ^{2} ( 6 )} \cdot 6^{\frac{x}{2}}) + C

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What is the integral of 2x6^{1/2 x} ?
The integral of 2x6^{1/2 x} is 8((3^{x/2}*2^{1/2 (x-2)}x)/(ln(6))-1/(ln^2(6))*6^{x/2})+C