What is the integral of (5^{2x}+3^{2x})/(3^{2x)} ?

The integral of (5^{2x}+3^{2x})/(3^{2x)} is (25^x)/(ln(25/9)*9^x)+x+C

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integral of (5^{2x}+3^{2x})/(3^{2x)}

\int \frac{5 ^{2 x} + 3 ^{2 x}}{3 ^{2 x}} d x

\frac{2 5 ^{x}}{ln ( \frac{25}{9} ) \cdot 9 ^{x}} + x + C

Solution steps

\int \frac{5 ^{2 x} + 3 ^{2 x}}{3 ^{2 x}} d x

Simplify

\frac{5 ^{2 x} + 3 ^{2 x}}{3 ^{2 x}} : \frac{2 5 ^{x} + 9 ^{x}}{9 ^{x}}

= \int \frac{2 5 ^{x} + 9 ^{x}}{9 ^{x}} d x

Expand

\frac{2 5 ^{x} + 9 ^{x}}{9 ^{x}} : (\frac{25}{9})^{x} + 1

= \int (\frac{25}{9})^{x} + 1 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 (\frac{25}{9})^{x} d x + \int 1 d x

\int (\frac{25}{9})^{x} d x = \frac{2 5 ^{x}}{ln ( \frac{25}{9} ) \cdot 9 ^{x}}

\int 1 d x = x

= \frac{2 5 ^{x}}{ln ( \frac{25}{9} ) \cdot 9 ^{x}} + x

Add a constant to the solution

= \frac{2 5 ^{x}}{ln ( \frac{25}{9} ) \cdot 9 ^{x}} + x + C

What is the integral of (5^{2x}+3^{2x})/(3^{2x)} ?
The integral of (5^{2x}+3^{2x})/(3^{2x)} is (25^x)/(ln(25/9)*9^x)+x+C