What is the integral of (5x)/(cos^2(x^2)) ?

The integral of (5x)/(cos^2(x^2)) is 5/2 tan(x^2)+C

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integral of (5x)/(cos^2(x^2))

\int \frac{5 x}{cos ^{2} ( x ^{2} )} d x

\frac{5}{2} tan (x^{2}) + C

Solution steps

\int \frac{5 x}{cos ^{2} ( x ^{2} )} d x

Take the constant out:

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

= 5 \cdot \int \frac{x}{cos ^{2} ( x ^{2} )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{cos ^{2} ( u )} d u = tan (u)

= 5 \cdot \frac{1}{2} tan (u)

Substitute back

u = x^{2}

= 5 \cdot \frac{1}{2} tan (x^{2})

Simplify

5 \cdot \frac{1}{2} tan (x^{2}) : \frac{5}{2} tan (x^{2})

= \frac{5}{2} tan (x^{2})

Add a constant to the solution

= \frac{5}{2} tan (x^{2}) + C

\int 21 d x

\frac{d ^{2} y}{d x ^{2}} = - 49 y

ln (tan (x^{3}))

\frac{d x}{d t} = - \frac{t}{x}

e^{2 t} sin (4 t)

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