What is the integral from 1 to sqrt(2 of)x2^{(x^2)} ?

The integral from 1 to sqrt(2 of)x2^{(x^2)} is 1/(ln(2))

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integral from 1 to sqrt(2 of)x2^{(x^2)}

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

\frac{1}{ln ( 2 )}

Decimal

1.44269 \dots

Solution steps

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

Apply u-substitution

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

Simplify

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

= \int_{1}^{2} 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} 2^{u} d u

Apply exponent rule:

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

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

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

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

Compute the boundaries:

\frac{2}{ln ( 2 )}

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

Simplify

= \frac{1}{ln ( 2 )}

What is the integral from 1 to sqrt(2 of)x2^{(x^2)} ?
The integral from 1 to sqrt(2 of)x2^{(x^2)} is 1/(ln(2))