What is the integral of (sqrt(x)sec(sqrt(x)))/(3x) ?

The integral of (sqrt(x)sec(sqrt(x)))/(3x) is 2/3 ln|tan(sqrt(x))+sec(sqrt(x))|+C

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integral of (sqrt(x)sec(sqrt(x)))/(3x)

\int \frac{x sec ( x )}{3 x} d x

\frac{2}{3} ln tan (x) + sec (x) + C

Solution steps

\int \frac{x sec ( x )}{3 x} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{x sec ( x )}{x} d x

Rewrite using trig identities

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{3} \cdot 2 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = x

= \frac{1}{3} \cdot 2 ln tan (x) + sec (x)

Simplify

\frac{1}{3} \cdot 2 ln tan (x) + sec (x) : \frac{2}{3} ln tan (x) + sec (x)

= \frac{2}{3} ln tan (x) + sec (x)

Add a constant to the solution

= \frac{2}{3} ln tan (x) + sec (x) + C

What is the integral of (sqrt(x)sec(sqrt(x)))/(3x) ?
The integral of (sqrt(x)sec(sqrt(x)))/(3x) is 2/3 ln|tan(sqrt(x))+sec(sqrt(x))|+C