What's the integral of $\int {cos}^{2} (x) {tan}^{3} (x) d x$ $int cos^2(x) tan^3(x) dx$ ?

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Answer 1 · 2018-04-02T15:16:33

$- ln | cos x | + \frac{1}{2} {cos}^{2} x + C$ $-lnabs(cosx)+1/2cos^2x+C$

${cos}^{2} (x) {tan}^{3} (x) = {cos}^{2} x \frac{{sin}^{3} x}{{cos}^{3} x}$ $cos^2(x) tan^3(x) = cos^2xsin^3x/cos^3x$

$= \frac{{sin}^{2} x}{cos x} sin x$ $= sin^2x/cosx sinx$

$= \frac{1 - {cos}^{2} x}{cos x} sin x$ $= (1-cos^2x)/cosx sinx$

$\int {cos}^{2} (x) {tan}^{3} (x) d x = \int (\frac{1}{cos x} + cos x) sin x d x$ $int cos^2(x) tan^3(x) dx = int (1/cosx + cosx) sinx dx$

$= - ln | cos x | + \frac{1}{2} {cos}^{2} x + C$ $= -lnabs(cosx)+1/2cos^2x+C$

What's the integral of int cos^2(x) tan^3(x) dx∫cos2(x)tan3(x)dxint cos^2(x) tan^3(x) dx?