Evaluate the definite integral by trigonometric integral. integral_{-pi / 2}^{pi / 2} (sin^2 x +...


Evaluate the definite integral by trigonometric integral.

{eq}\displaystyle \int\limits_{-\dfrac \pi 2}^{\dfrac \pi 2} (\sin^2 x + 1)\ dx {/eq}

Definite Integration:

The definite integral of a function gives the area bounded by the function and the horizontal axis. In solving the integral of a trigonometric function, the knowledge of trigonometric ratio is very crucial. It is recommended that a student must familiarize themselves in manipulating a trigonometric function by applying trigonometric identities. In solving the given integral, the following formula will be applied:

$$\begin{align} 1+\cos 2\theta&= 2\cos^2 \theta\\[0.3cm] 1-\cos 2\theta &= 2\sin^2\theta\\[0.3cm] \int \cos (ax)&=\frac{\sin (ax) }{a}+C\\[0.3cm] \int \sin (ax)&=-\frac{\cos (ax) }{a}+C\\[0.3cm] \int x^n\, \text{d}x&=\frac { x^{n+1}} {n+1}+C & \left[\text{ Power rule of integration } \right]\\[0.3cm] \end{align} $$

Answer and Explanation: 1

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We have the following given data

$$\begin{align} \int\limits_{-\dfrac \pi 2}^{\dfrac \pi 2} (\sin^2 x + 1)\ \text{d}x &=? \end{align} \\ $$


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Definite Integrals: Definition


Chapter 12 / Lesson 6

A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.

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