What is the integral of (7sec^2(2x))/(3-5tan(2x)) ?

The integral of (7sec^2(2x))/(3-5tan(2x)) is -7/10 ln|3-5tan(2x)|+C

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integral of (7sec^2(2x))/(3-5tan(2x))

\int \frac{7 sec ^{2} ( 2 x )}{3 - 5 tan ( 2 x )} d x

- \frac{7}{10} ln ∣ 3 - 5 tan (2 x) ∣ + C

Solution steps

\int \frac{7 sec ^{2} ( 2 x )}{3 - 5 tan ( 2 x )} d x

Take the constant out:

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

= 7 \cdot \int \frac{sec ^{2} ( 2 x )}{3 - 5 tan ( 2 x )} d x

Apply u-substitution

= 7 \cdot \int - \frac{1}{10 u} d u

Take the constant out:

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

= 7 (- \frac{1}{10} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 (- \frac{1}{10} ln ∣ u ∣)

Substitute back

u = 3 - 5 tan (2 x)

= 7 (- \frac{1}{10} ln ∣ 3 - 5 tan (2 x) ∣)

Simplify

7 (- \frac{1}{10} ln ∣ 3 - 5 tan (2 x) ∣) : - \frac{7}{10} ln ∣ 3 - 5 tan (2 x) ∣

= - \frac{7}{10} ln ∣ 3 - 5 tan (2 x) ∣

Add a constant to the solution

= - \frac{7}{10} ln ∣ 3 - 5 tan (2 x) ∣ + C

\int arcsinh (x) d x

x \to + 0 + lim (x^{12} ln (x))

\frac{d}{d x} (\frac{2}{2 x ^{4} - 5})

s l o p e (- 3.7) (0.3)

\frac{d}{d x} (3 \cdot 4^{x^{2} + 2})

What is the integral of (7sec^2(2x))/(3-5tan(2x)) ?
The integral of (7sec^2(2x))/(3-5tan(2x)) is -7/10 ln|3-5tan(2x)|+C