What is the integral of 3x2^{1/2 x} ?

The integral of 3x2^{1/2 x} is (12)/(ln^2(2))(ln(2)*2^{1/2 (x-2)}x-2^{1/2 x})+C

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integral of 3x2^{1/2 x}

\int 3 x 2^{\frac{1}{2} x} d x

\frac{12}{ln ^{2} ( 2 )} (ln (2) \cdot 2^{\frac{1}{2} (x - 2)} x - 2^{\frac{1}{2} x}) + C

Solution steps

\int 3 x \cdot 2^{\frac{1}{2} x} d x

Take the constant out:

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

= 3 \cdot \int x \cdot 2^{\frac{1}{2} x} d x

Apply u-substitution

: \int \frac{4 ln ( u )}{ln ^{2} ( 2 )} d u

= 3 \cdot \int \frac{4 ln ( u )}{ln ^{2} ( 2 )} d u

Take the constant out:

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

= 3 \cdot \frac{4}{ln ^{2} ( 2 )} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= 3 \cdot \frac{4}{ln ^{2} ( 2 )} (u ln (u) - \int 1 d u)

\int 1 d u = u

= 3 \cdot \frac{4}{ln ^{2} ( 2 )} (u ln (u) - u)

Substitute back

u = 2^{\frac{1}{2} x}

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

Simplify

3 \cdot \frac{4}{ln ^{2} ( 2 )} (2^{\frac{1}{2} x} ln (2^{\frac{1}{2} x}) - 2^{\frac{1}{2} x}) : \frac{12}{ln ^{2} ( 2 )} (ln (2) \cdot 2^{\frac{1}{2} (x - 2)} x - 2^{\frac{1}{2} x})

= \frac{12}{ln ^{2} ( 2 )} (ln (2) \cdot 2^{\frac{1}{2} (x - 2)} x - 2^{\frac{1}{2} x})

Add a constant to the solution

= \frac{12}{ln ^{2} ( 2 )} (ln (2) \cdot 2^{\frac{1}{2} (x - 2)} x - 2^{\frac{1}{2} x}) + C

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What is the integral of 3x2^{1/2 x} ?
The integral of 3x2^{1/2 x} is (12)/(ln^2(2))(ln(2)*2^{1/2 (x-2)}x-2^{1/2 x})+C