What is the integral of (tan(5x))/(cos^2(5x)) ?

The integral of (tan(5x))/(cos^2(5x)) is 1/10 tan^2(5x)+C

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integral of (tan(5x))/(cos^2(5x))

\int \frac{tan ( 5 x )}{cos ^{2} ( 5 x )} d x

\frac{1}{10} tan^{2} (5 x) + C

Solution steps

\int \frac{tan ( 5 x )}{cos ^{2} ( 5 x )} d x

Use the basic trigonometric identity:

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

= \int sec^{2} (5 x) tan (5 x) d x

Apply u-substitution

= \int \frac{u}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u d u

Apply the Power Rule

= \frac{1}{5} \cdot \frac{u ^{2}}{2}

Substitute back

u = tan (5 x)

= \frac{1}{5} \cdot \frac{tan ^{2} ( 5 x )}{2}

Simplify

\frac{1}{5} \cdot \frac{tan ^{2} ( 5 x )}{2} : \frac{1}{10} tan^{2} (5 x)

= \frac{1}{10} tan^{2} (5 x)

Add a constant to the solution

= \frac{1}{10} tan^{2} (5 x) + C

What is the integral of (tan(5x))/(cos^2(5x)) ?
The integral of (tan(5x))/(cos^2(5x)) is 1/10 tan^2(5x)+C