What is the integral of 3/(cos^2(5x)) ?

The integral of 3/(cos^2(5x)) is 3/5 tan(5x)+C

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integral of 3/(cos^2(5x))

\int \frac{3}{cos ^{2} ( 5 x )} d x

\frac{3}{5} tan (5 x) + C

Solution steps

\int \frac{3}{cos ^{2} ( 5 x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{cos ^{2} ( 5 x )} d x

Rewrite using trig identities

= 3 \cdot \int sec^{2} (5 x) d x

Apply u-substitution

: \int sec^{2} (u) \frac{1}{5} d u

= 3 \cdot \int sec^{2} (u) \frac{1}{5} d u

Take the constant out:

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

= 3 \cdot \frac{1}{5} \cdot \int sec^{2} (u) d u

Use the common integral:

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

= 3 \cdot \frac{1}{5} tan (u)

Substitute back

u = 5 x

= 3 \cdot \frac{1}{5} tan (5 x)

Simplify

3 \cdot \frac{1}{5} tan (5 x) : \frac{3}{5} tan (5 x)

= \frac{3}{5} tan (5 x)

Add a constant to the solution

= \frac{3}{5} tan (5 x) + C

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What is the integral of 3/(cos^2(5x)) ?
The integral of 3/(cos^2(5x)) is 3/5 tan(5x)+C