What is the integral of sec^4(x)sqrt(tan(x)) ?

The integral of sec^4(x)sqrt(tan(x)) is 2/3 tan^{3/2}(x)+2/7 tan^{7/2}(x)+C

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integral of sec^4(x)sqrt(tan(x))

\int sec^{4} (x) tan (x) d x

\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x) + C

Solution steps

\int sec^{4} (x) tan (x) d x

Rewrite using trig identities

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

Apply u-substitution

= \int u (1 + u^{2}) d u

Expand

u (1 + u^{2}) : u + u^{\frac{5}{2}}

= \int u + u^{\frac{5}{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u + \int u^{\frac{5}{2}} d u

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= \frac{2}{3} u^{\frac{3}{2}} + \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = tan (x)

= \frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x)

Add a constant to the solution

= \frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x) + C

What is the integral of sec^4(x)sqrt(tan(x)) ?
The integral of sec^4(x)sqrt(tan(x)) is 2/3 tan^{3/2}(x)+2/7 tan^{7/2}(x)+C