What is the integral from 0 to 2 of (x)2^{4x^2} ?

The integral from 0 to 2 of (x)2^{4x^2} is (65535)/(8ln(2))

integral from 0 to 2 of (x)2^{4x^2}

\int_{0}^{2} (x) 2^{4 x^{2}} d x

\frac{65535}{8 ln ( 2 )}

Decimal

11818.37743 \dots

Solution steps

\int_{0}^{2} x \cdot 2^{4 x^{2}} d x

Apply u-substitution

= \int_{0}^{16} 2^{u - 3} d u

Simplify

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

= \int_{0}^{16} 2^{u} \cdot 2^{- 3} d u

Take the constant out:

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

= 2^{- 3} \cdot \int_{0}^{16} 2^{u} d u

2^{- 3} = \frac{1}{8}

= \frac{1}{8} \cdot \int_{0}^{16} 2^{u} d u

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

= \frac{1}{8} [\frac{2 ^{u}}{ln ( 2 )}]_{0}^{16}

Compute the boundaries:

\frac{65535}{ln ( 2 )}

= \frac{1}{8} \cdot \frac{65535}{ln ( 2 )}

Simplify

= \frac{65535}{8 ln ( 2 )}

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What is the integral from 0 to 2 of (x)2^{4x^2} ?
The integral from 0 to 2 of (x)2^{4x^2} is (65535)/(8ln(2))