What is the integral of (tan(x))/(cos^3(x)) ?

The integral of (tan(x))/(cos^3(x)) is 1/3 sec^3(x)+C

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integral of (tan(x))/(cos^3(x))

\int \frac{tan ( x )}{cos ^{3} ( x )} d x

\frac{1}{3} sec^{3} (x) + C

Solution steps

\int \frac{tan ( x )}{cos ^{3} ( x )} d x

Use the basic trigonometric identity:

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

= \int sec^{3} (x) tan (x) d x

Apply u-substitution

: \int \frac{1}{3} d u

= \int \frac{1}{3} d u

Integral of a constant:

\int a d x = a x

= \frac{1}{3} u

Substitute back

u = sec^{3} (x)

= \frac{1}{3} sec^{3} (x)

Add a constant to the solution

= \frac{1}{3} sec^{3} (x) + C

\frac{d y}{y ^{2} - 4} = d x

\frac{\partial}{\partial x} (- 3 x)

\int \frac{1}{x ^{2}} - \frac{1}{x x} + 5 d x

\frac{d y}{d x} = 3 y e^{x + 7}

\int - \frac{cos ( x )}{x} d x

What is the integral of (tan(x))/(cos^3(x)) ?
The integral of (tan(x))/(cos^3(x)) is 1/3 sec^3(x)+C