What is the integral from x to y of 2^{ln(t)} ?

The integral from x to y of 2^{ln(t)} is ((2e)^{ln(y)}-(2e)^{ln(x)})/(ln(2e))

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integral from x to y of 2^{ln(t)}

\int_{x}^{y} 2^{l n (t)} d t

\frac{( 2 e ) ^{l n (y)} - ( 2 e ) ^{l n (x)}}{ln ( 2 e )}

Solution steps

\int_{x}^{y} 2^{l n (t)} d t

Apply u-substitution

: \int_{l n (x)}^{l n (y)} 2^{u} e^{u} d u

= \int_{l n (x)}^{l n (y)} 2^{u} e^{u} d u

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

2^{u} e^{u} = (2 e)^{u}

= \int_{l n (x)}^{l n (y)} (2 e)^{u} d u

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

= [\frac{( 2 e ) ^{u}}{ln ( 2 e )}]_{l n (x)}^{l n (y)}

Compute the boundaries:

\frac{( 2 e ) ^{l n (y)}}{ln ( 2 e )} - \frac{( 2 e ) ^{l n (x)}}{ln ( 2 e )}

= \frac{( 2 e ) ^{l n (y)}}{ln ( 2 e )} - \frac{( 2 e ) ^{l n (x)}}{ln ( 2 e )}

Simplify

= \frac{( 2 e ) ^{l n (y)} - ( 2 e ) ^{l n (x)}}{ln ( 2 e )}

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What is the integral from x to y of 2^{ln(t)} ?
The integral from x to y of 2^{ln(t)} is ((2e)^{ln(y)}-(2e)^{ln(x)})/(ln(2e))