What is the integral of 1/3 sec^2(8x) ?

The integral of 1/3 sec^2(8x) is 1/24 tan(8x)+C

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integral of 1/3 sec^2(8x)

\int \frac{1}{3} sec^{2} (8 x) d x

\frac{1}{24} tan (8 x) + C

Solution steps

\int \frac{1}{3} sec^{2} (8 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 sec^{2} (8 x) d x

Apply u-substitution

= \frac{1}{3} \cdot \int sec^{2} (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{8} \cdot \int sec^{2} (u) d u

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{8} tan (u)

Substitute back

u = 8 x

= \frac{1}{3} \cdot \frac{1}{8} tan (8 x)

Simplify

\frac{1}{3} \cdot \frac{1}{8} tan (8 x) : \frac{1}{24} tan (8 x)

= \frac{1}{24} tan (8 x)

Add a constant to the solution

= \frac{1}{24} tan (8 x) + C

What is the integral of 1/3 sec^2(8x) ?
The integral of 1/3 sec^2(8x) is 1/24 tan(8x)+C