What is the integral from 0 to pi/3 of (cos(x)+sec(x))^2 ?

The integral from 0 to pi/3 of (cos(x)+sec(x))^2 is sqrt(3)+(20pi+3sqrt(3))/(24)

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integral from 0 to pi/3 of (cos(x)+sec(x))^2

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

3 + \frac{20 π + 3 3}{24}

Decimal Notation

4.56655 \dots

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( cos ^{2} ( x ) + 1 ) ^{2}}{cos ^{2} ( x )} : cos^{2} (x) + 2 + \frac{1}{cos ^{2} ( x )}

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

Apply the Sum Rule:

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

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

\int_{0}^{\frac{π}{3}} cos^{2} (x) d x = \frac{4 π + 3 3}{24}

\int_{0}^{\frac{π}{3}} 2 d x = \frac{2 π}{3}

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

= \frac{4 π + 3 3}{24} + \frac{2 π}{3} + 3

Simplify

\frac{4 π + 3 3}{24} + \frac{2 π}{3} + 3 : 3 + \frac{20 π + 3 3}{24}

= 3 + \frac{20 π + 3 3}{24}

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What is the integral from 0 to pi/3 of (cos(x)+sec(x))^2 ?
The integral from 0 to pi/3 of (cos(x)+sec(x))^2 is sqrt(3)+(20pi+3sqrt(3))/(24)