What is the integral of (2(tan(x)+3)^7)/(cos^2(x)) ?

The integral of (2(tan(x)+3)^7)/(cos^2(x)) is 1/4 (tan(x)+3)^8+C

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integral of (2(tan(x)+3)^7)/(cos^2(x))

\int \frac{2 ( tan ( x ) + 3 ) ^{7}}{cos ^{2} ( x )} d x

\frac{1}{4} (tan (x) + 3)^{8} + C

Solution steps

\int \frac{2 ( tan ( x ) + 3 ) ^{7}}{cos ^{2} ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{( tan ( x ) + 3 ) ^{7}}{cos ^{2} ( x )} d x

Use the basic trigonometric identity:

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

= 2 \cdot \int sec^{2} (x) (tan (x) + 3)^{7} d x

Apply u-substitution

: \int u^{7} d u

= 2 \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= 2 \cdot \frac{u ^{8}}{8}

Substitute back

u = tan (x) + 3

= 2 \cdot \frac{( tan ( x ) + 3 ) ^{8}}{8}

Simplify

2 \cdot \frac{( tan ( x ) + 3 ) ^{8}}{8} : \frac{1}{4} (tan (x) + 3)^{8}

= \frac{1}{4} (tan (x) + 3)^{8}

Add a constant to the solution

= \frac{1}{4} (tan (x) + 3)^{8} + C

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What is the integral of (2(tan(x)+3)^7)/(cos^2(x)) ?
The integral of (2(tan(x)+3)^7)/(cos^2(x)) is 1/4 (tan(x)+3)^8+C