What is the integral from 1 to sqrt(2 of)x6^{(x^2)} ?

The integral from 1 to sqrt(2 of)x6^{(x^2)} is (15)/(ln(6))

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integral from 1 to sqrt(2 of)x6^{(x^2)}

\int_{1}^{2} x 6^{(x^{2})} d x

\frac{15}{ln ( 6 )}

Solution steps

\int_{1}^{2} x \cdot 6^{x^{2}} d x

Apply u-substitution

: \int_{1}^{2} 3^{u} \cdot 2^{u - 1} d u

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

Simplify

2^{u - 1} : 2^{u} \cdot 2^{- 1}

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

Take the constant out:

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

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

Apply exponent rule:

a^{- 1} = \frac{1}{a}

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

Simplify

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

= \frac{1}{2} \cdot \int_{1}^{2} 6^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{2} [\frac{6 ^{u}}{ln ( 6 )}]_{1}^{2}

Compute the boundaries:

\frac{30}{ln ( 6 )}

= \frac{1}{2} \cdot \frac{30}{ln ( 6 )}

Simplify

= \frac{15}{ln ( 6 )}

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What is the integral from 1 to sqrt(2 of)x6^{(x^2)} ?
The integral from 1 to sqrt(2 of)x6^{(x^2)} is (15)/(ln(6))