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

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

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integral of (a^{tan(m)})/(cos^2(m))

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

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

Solution steps

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

Use the basic trigonometric identity:

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

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

Apply u-substitution

: \int a^{u} d u

= \int a^{u} d u

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

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

Substitute back

u = tan (m)

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

Add a constant to the solution

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

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