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Prove that for positive​ integers, Integral from nothing to nothing tangent Superscript n Baseline x dx...

Prove that for positive​ integers,

Integral from nothing to nothing tangent Superscript n Baseline x dx equals StartFraction tangent Superscript n minus 1 Baseline x Over n minus 1 EndFraction minus Integral from nothing to nothing tangent Superscript n minus 2 Baseline x dx comma n not equals 1∫tannx dx=tann−1xn−1−∫tann−2x dx, n≠1.

Use the formula to evaluate

Integral from 0 to StartFraction pi Over 3 EndFraction 4 tangent Superscript 5 Baseline x dx∫0π34tan5x dx.

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