What is the integral of (5^{tan(x)})/(cos^2(x)) ?

The integral of (5^{tan(x)})/(cos^2(x)) is (5^{tan(x)})/(ln(5))+C

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integral of (5^{tan(x)})/(cos^2(x))

\int \frac{5 ^{t a n (x)}}{cos ^{2} ( x )} d x

\frac{5 ^{t a n (x)}}{ln ( 5 )} + C

Solution steps

\int \frac{5 ^{t a n (x)}}{cos ^{2} ( x )} d x

Rewrite using trig identities

= \int 5^{t a n (x)} sec^{2} (x) d x

Apply u-substitution

: \int 5^{u} d u

= \int 5^{u} d u

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

= \frac{5 ^{u}}{ln ( 5 )}

Substitute back

u = tan (x)

= \frac{5 ^{t a n (x)}}{ln ( 5 )}

Add a constant to the solution

= \frac{5 ^{t a n (x)}}{ln ( 5 )} + C

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What is the integral of (5^{tan(x)})/(cos^2(x)) ?
The integral of (5^{tan(x)})/(cos^2(x)) is (5^{tan(x)})/(ln(5))+C