What is the integral of sqrt(tan(7x))(sec(7x))^2 ?

The integral of sqrt(tan(7x))(sec(7x))^2 is 2/21 tan^{3/2}(7x)+C

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integral of sqrt(tan(7x))(sec(7x))^2

\int tan (7 x) (sec (7 x))^{2} d x

\frac{2}{21} tan^{\frac{3}{2}} (7 x) + C

Solution steps

\int tan (7 x) (sec (7 x))^{2} d x

Rewrite using trig identities

= \int \frac{sin ( 7 x )}{cos ( 7 x )} sec^{2} (7 x) d x

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

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

Apply u-substitution

: \int \frac{2 u ^{2}}{7} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

= \frac{2}{7} \cdot \frac{u ^{3}}{3}

Substitute back

u = tan (7 x)

= \frac{2}{7} \cdot \frac{( tan ( 7 x ) ) ^{3}}{3}

Simplify

\frac{2}{7} \cdot \frac{( tan ( 7 x ) ) ^{3}}{3} : \frac{2}{21} tan^{\frac{3}{2}} (7 x)

= \frac{2}{21} tan^{\frac{3}{2}} (7 x)

Add a constant to the solution

= \frac{2}{21} tan^{\frac{3}{2}} (7 x) + C

x \to 0 lim (\frac{3 x - 2}{9 x + 7})

\int_{0}^{x} e^{x^{2}} d x

\int - 4 t e^{- 4 t} d t

\int 5 tan (x) d x

4 y^{^{''}} + 3 y = 0

What is the integral of sqrt(tan(7x))(sec(7x))^2 ?
The integral of sqrt(tan(7x))(sec(7x))^2 is 2/21 tan^{3/2}(7x)+C