What is the integral from 0 to pi/2 of (1+sin(x))^2 ?

The integral from 0 to pi/2 of (1+sin(x))^2 is (3pi)/4+2

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integral from 0 to pi/2 of (1+sin(x))^2

\int_{0}^{\frac{π}{2}} (1 + sin (x))^{2} d x

\frac{3 π}{4} + 2

Decimal

4.35619 \dots

Solution steps

\int_{0}^{\frac{π}{2}} (1 + sin (x))^{2} d x

Expand

(1 + sin (x))^{2} : 1 + 2 sin (x) + sin^{2} (x)

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

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

\int_{0}^{\frac{π}{2}} 2 sin (x) d x = 2

\int_{0}^{\frac{π}{2}} sin^{2} (x) d x = \frac{π}{4}

= \frac{π}{2} + 2 + \frac{π}{4}

Simplify

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

= \frac{3 π}{4} + 2

What is the integral from 0 to pi/2 of (1+sin(x))^2 ?
The integral from 0 to pi/2 of (1+sin(x))^2 is (3pi)/4+2