What is the integral of 5sec^2(1+5x) ?

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

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integral of 5sec^2(1+5x)

\int 5 sec^{2} (1 + 5 x) d x

tan (1 + 5 x) + C

Solution steps

\int 5 sec^{2} (1 + 5 x) d x

Take the constant out:

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

= 5 \cdot \int sec^{2} (1 + 5 x) d x

Apply u-substitution

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

= 5 \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

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

Use the common integral:

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

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

Substitute back

u = 1 + 5 x

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

Simplify

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

= tan (1 + 5 x)

Add a constant to the solution

= tan (1 + 5 x) + C

\frac{1}{x ^{8}} \frac{d y}{d x} = 12 + 15 y

\frac{5}{ln ( x )}

\frac{d}{d x} (\frac{1}{x - 2} - 3)

t \to 1 lim (cos (π t))

y = x^{3}, y = 4 + 2 x, x = 0

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