What is the integral of ((2+tan(x))^8)/(cos^2(x)) ?

The integral of ((2+tan(x))^8)/(cos^2(x)) is ((2+tan(x))^9)/9+C

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integral of ((2+tan(x))^8)/(cos^2(x))

\int \frac{( 2 + tan ( x ) ) ^{8}}{cos ^{2} ( x )} d x

\frac{( 2 + tan ( x ) ) ^{9}}{9} + C

Solution steps

\int \frac{( 2 + tan ( x ) ) ^{8}}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

= \int sec^{2} (x) (2 + tan (x))^{8} d x

Apply u-substitution

: \int u^{8} d u

= \int u^{8} d u

Apply the Power Rule

: \frac{u ^{9}}{9}

= \frac{u ^{9}}{9}

Substitute back

u = 2 + tan (x)

= \frac{( 2 + tan ( x ) ) ^{9}}{9}

Add a constant to the solution

= \frac{( 2 + tan ( x ) ) ^{9}}{9} + C

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What is the integral of ((2+tan(x))^8)/(cos^2(x)) ?
The integral of ((2+tan(x))^8)/(cos^2(x)) is ((2+tan(x))^9)/9+C