What is the integral of (e^{aln(x)}+e^{xln(a)}) ?

The integral of (e^{aln(x)}+e^{xln(a)}) is (x^{a+1})/(a+1)+(a^x)/(ln(a))+C

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integral of (e^{aln(x)}+e^{xln(a)})

\int (e^{a l n (x)} + e^{x l n (a)}) d x

\frac{x ^{a + 1}}{a + 1} + \frac{a ^{x}}{ln ( a )} + C

Solution steps

\int e^{a l n (x)} + e^{x l n (a)} d x

Simplify

e^{a l n (x)} + e^{x l n (a)} : x^{a} + a^{x}

= \int x^{a} + a^{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{a} d x + \int a^{x} d x

\int x^{a} d x = \frac{x ^{a + 1}}{a + 1}

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

= \frac{x ^{a + 1}}{a + 1} + \frac{a ^{x}}{ln ( a )}

Add a constant to the solution

= \frac{x ^{a + 1}}{a + 1} + \frac{a ^{x}}{ln ( a )} + C

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What is the integral of (e^{aln(x)}+e^{xln(a)}) ?
The integral of (e^{aln(x)}+e^{xln(a)}) is (x^{a+1})/(a+1)+(a^x)/(ln(a))+C