What is the integral of (1+tan(x))/(cos^2(x)) ?

The integral of (1+tan(x))/(cos^2(x)) is ((1+tan(x))^2)/2+C

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integral of (1+tan(x))/(cos^2(x))

\int \frac{1 + tan ( x )}{cos ^{2} ( x )} d x

\frac{( 1 + tan ( x ) ) ^{2}}{2} + C

Solution steps

\int \frac{1 + tan ( x )}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

= \int sec^{2} (x) (1 + tan (x)) d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{u ^{2}}{2}

Substitute back

u = 1 + tan (x)

= \frac{( 1 + tan ( x ) ) ^{2}}{2}

Add a constant to the solution

= \frac{( 1 + tan ( x ) ) ^{2}}{2} + C

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What is the integral of (1+tan(x))/(cos^2(x)) ?
The integral of (1+tan(x))/(cos^2(x)) is ((1+tan(x))^2)/2+C