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

The integral of ((1+tan(x))^3)/(cos^2(x)) is ((1+tan(x))^4)/4+C

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

\int \frac{( 1 + tan ( x ) ) ^{3}}{cos ^{2} ( x )} d x

\frac{( 1 + tan ( x ) ) ^{4}}{4} + C

Solution steps

\int \frac{( 1 + tan ( x ) ) ^{3}}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

= \int sec^{2} (x) (1 + tan (x))^{3} d x

Apply u-substitution

: \int u^{3} d u

= \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{u ^{4}}{4}

Substitute back

u = 1 + tan (x)

= \frac{( 1 + tan ( x ) ) ^{4}}{4}

Add a constant to the solution

= \frac{( 1 + tan ( x ) ) ^{4}}{4} + C

\int (\frac{2}{1 + x ^{2}} - \frac{3}{1 - x ^{2}}) d x

\int (cos (x) + 4^{x}) d x

\int \frac{x ^{- 1}}{- 1} d x

\int \frac{( a ^{x} ln ( a ) )}{( a ^{x} + 1 )} d x

\int \frac{8}{( x ^{2} - 4 )} d x

What is the integral of ((1+tan(x))^3)/(cos^2(x)) ?
The integral of ((1+tan(x))^3)/(cos^2(x)) is ((1+tan(x))^4)/4+C