What is the integral of sec^2(5t) ?

The integral of sec^2(5t) is 1/5 tan(5t)+C

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integral of sec^2(5t)

\int sec^{2} (5 t) d t

\frac{1}{5} tan (5 t) + C

Solution steps

\int sec^{2} (5 t) d t

Apply u-substitution

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

= \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

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

Use the common integral:

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

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

Substitute back

u = 5 t

= \frac{1}{5} tan (5 t)

Add a constant to the solution

= \frac{1}{5} tan (5 t) + C

g (x) = x^{2} (1 - 5 x)

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