`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`

Let I = `int sqrt(x)*sec(x^(3/2))*tan(x^(3/2))"d"x` Put `x^(3/2)` = t &there4; `3/2x^(1/2) "d"x` = dt &there4; `sqrt(x) "d"x = 2/3 "dt"` &there4; I = `2/3 int sec "t"*tan"t"* "dt"` = `2/3 sec "t" + "c"` &there4; I = `2/3 sec(x^(3/2)) + "c"`

∫x sec(x)32tan(x)32dx - Mathematics and Statistics

Question

$\int \sqrt{x} {\sec (x)}^{\frac{3}{2}} {\tan (x)}^{\frac{3}{2}} d x$

Sum

Solution

Let I = $\int \sqrt{x} \cdot \sec (x^{\frac{3}{2}}) \cdot \tan (x^{\frac{3}{2}}) d x$

Put $x^{\frac{3}{2}}$ = t

∴ $\frac{3}{2} x^{\frac{1}{2}} d x$ = dt

∴ $\sqrt{x} d x = \frac{2}{3} dt$

∴ I = $\frac{2}{3} \int \sec t \cdot \tan t \cdot dt$

= $\frac{2}{3} \sec t + c$

∴ I = $\frac{2}{3} \sec (x^{\frac{3}{2}}) + c$

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Methods of Integration: Integration by Substitution

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Chapter 2.3: Indefinite Integration - Very Short Answers

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