What is the integral of (1+2e^{tan(y)})sec^2(y) ?

The integral of (1+2e^{tan(y)})sec^2(y) is tan(y)+2e^{tan(y)}+C

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integral of (1+2e^{tan(y)})sec^2(y)

\int (1 + 2 e^{t a n (y)}) sec^{2} (y) d y

tan (y) + 2 e^{t a n (y)} + C

Solution steps

\int (1 + 2 e^{t a n (y)}) sec^{2} (y) d y

Rewrite using trig identities

= \int \frac{1 + 2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )} d y

Expand

\frac{1 + 2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )} : \frac{1}{cos ^{2} ( y )} + \frac{2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )}

= \int \frac{1}{cos ^{2} ( y )} + \frac{2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )} d y

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{1}{cos ^{2} ( y )} d y + \int \frac{2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )} d y

\int \frac{1}{cos ^{2} ( y )} d y = tan (y)

\int \frac{2 e ^{\frac{s i n ( y )}{c o s ( y )}}}{cos ^{2} ( y )} d y = 2 e^{t a n (y)}

= tan (y) + 2 e^{t a n (y)}

Add a constant to the solution

= tan (y) + 2 e^{t a n (y)} + C

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What is the integral of (1+2e^{tan(y)})sec^2(y) ?
The integral of (1+2e^{tan(y)})sec^2(y) is tan(y)+2e^{tan(y)}+C