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

The integral of ((1+tan(x))^2)/(cos^2(x)) is ((1+tan(x))^3)/3+C

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

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

\frac{( 1 + tan ( x ) ) ^{3}}{3} + C

Solution steps

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

Use the basic trigonometric identity:

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

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{u ^{3}}{3}

Substitute back

u = 1 + tan (x)

= \frac{( 1 + tan ( x ) ) ^{3}}{3}

Add a constant to the solution

= \frac{( 1 + tan ( x ) ) ^{3}}{3} + C

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What is the integral of ((1+tan(x))^2)/(cos^2(x)) ?
The integral of ((1+tan(x))^2)/(cos^2(x)) is ((1+tan(x))^3)/3+C