What is the integral of sec^7(3x)tan(3x) ?

The integral of sec^7(3x)tan(3x) is 1/21 sec^7(3x)+C

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integral of sec^7(3x)tan(3x)

\int sec^{7} (3 x) tan (3 x) d x

\frac{1}{21} sec^{7} (3 x) + C

Solution steps

\int sec^{7} (3 x) tan (3 x) d x

Apply u-substitution

: \int sec^{7} (u) tan (u) \frac{1}{3} d u

= \int sec^{7} (u) tan (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sec^{7} (u) tan (u) d u

Apply u-substitution

: \int v^{6} d v

= \frac{1}{3} \cdot \int v^{6} d v

Apply the Power Rule

: \frac{v ^{7}}{7}

= \frac{1}{3} \cdot \frac{v ^{7}}{7}

Substitute back

= \frac{1}{3} \cdot \frac{sec ^{7} ( 3 x )}{7}

Simplify

\frac{1}{3} \cdot \frac{sec ^{7} ( 3 x )}{7} : \frac{1}{21} sec^{7} (3 x)

= \frac{1}{21} sec^{7} (3 x)

Add a constant to the solution

= \frac{1}{21} sec^{7} (3 x) + C

x \to 0 - lim (cot (x))

(x (csc (\frac{y}{x})) - y) d x + x d y = 0

f (x) = (2 x)^{3}

y = \frac{1}{( x - 1 )}

\frac{\partial}{\partial x} (\frac{1}{x ^{2}} + y)

What is the integral of sec^7(3x)tan(3x) ?
The integral of sec^7(3x)tan(3x) is 1/21 sec^7(3x)+C