What is the integral of sec^2(sqrt(x)) ?

The integral of sec^2(sqrt(x)) is 2(sqrt(x)tan(sqrt(x))+ln|cos(sqrt(x))|)+C

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integral of sec^2(sqrt(x))

\int sec^{2} (x) d x

2 (x tan (x) + ln cos (x)) + C

Solution steps

\int sec^{2} (x) d x

Apply u-substitution

: \int sec^{2} (u) \cdot 2 u d u

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

Take the constant out:

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

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

Apply Integration By Parts

: u tan (u) - \int tan (u) d u

= 2 (u tan (u) - \int tan (u) d u)

\int tan (u) d u = - ln ∣ cos (u) ∣

= 2 (u tan (u) - (- ln ∣ cos (u) ∣))

Substitute back

u = x

= 2 (x tan (x) - (- ln cos (x)))

Simplify

= 2 (x tan (x) + ln cos (x))

Add a constant to the solution

= 2 (x tan (x) + ln cos (x)) + C

y^{^{''}} + 1.5 y^{^{'}} + 0.037 y = 0, y (0) = 2, y^{^{'}} (0) = 1

\frac{d}{d x} (3^{x} + 5)

\frac{x}{x ^{2} + 25}

\int \frac{- 2 ln ( x )}{x} d x

\int x^{2} 0. 4^{- x} d x

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