Expert Answer to "Evaluate the definite integral. ∫0π3tan2xdx"

jernplate8

Answered question

2021-10-16

Evaluate the definite integral.

\int_{0}^{\frac{π}{3}} \tan^{2} x d x

krolaniaN

Skilled2021-10-17Added 86 answers

Step 1
The given integral is

\int_{0}^{\frac{π}{3}} \tan^{2} x d x

.
Step 2

\int_{0}^{\frac{π}{3}} \tan^{2} x d x = \int_{0}^{\frac{π}{3}} (1) d x + \int_{0}^{\frac{π}{3}} (\sec^{2} x) d x

= - \int_{0}^{\frac{π}{3}} (1) d x + \int_{0}^{\frac{π}{3}} (\sec^{2} x) d x

= {[- x + \tan x]}_{0}^{\frac{π}{3}}

= (- \frac{π}{3} + \tan \frac{π}{3}) - (- 0 + \tan 0)

= - \frac{π}{3} + \sqrt{3}

Hence, the value of the integral

\int_{0}^{\frac{π}{3}} \tan^{2} x d x i s - \frac{π}{3} + \sqrt{3}

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