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

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

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

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

\frac{2 ^{t a n (x)}}{ln ( 2 )} + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Apply u-substitution

= \int 2^{u} d u

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

= \frac{2 ^{u}}{ln ( 2 )}

Substitute back

u = tan (x)

= \frac{2 ^{t a n (x)}}{ln ( 2 )}

Add a constant to the solution

= \frac{2 ^{t a n (x)}}{ln ( 2 )} + C

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