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

The integral of x/((cos(x^2))^2) is 1/2 tan(x^2)+C

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

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

\frac{1}{2} tan (x^{2}) + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x^{2}

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

Add a constant to the solution

= \frac{1}{2} tan (x^{2}) + C

\int - 5 x^{2} (3 x - 8)^{4} d x

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\int \frac{1}{1 + \frac{1}{x ^{2}}} d x

\int \frac{sec ^{2} ( 5 x )}{1 - tan ( 5 x )} d x

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

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