Chapter 5: Q 78. (page 429)

Solve the definite integration.
$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{- 1} x d x$

Short Answer

Expert verified

The solution is $0$ .

Step by step solution

Step 1. Given information.

The given integral is $\int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{- 1} x d x$ .

Step 2. First, solve the indefinite integral.

$\begin{array}{rcl} I & = & \int \sin^{- 1} x d x \\ = & \sin^{- 1} x \int d x - \int [\frac{d}{d x} (\sin^{- 1} x) \int d x] d x \\ = & \sin^{- 1} x \cdot x - \int \frac{1}{\sqrt{1 - x^{2}}} \cdot x d x \\ = & \sin^{- 1} x \cdot x - \int \frac{x}{\sqrt{1 - x^{2}}} d x \\ = & x \sin^{- 1} x - I_{1} \end{array}$

Step 3. Solve for I1.

$I_{1} = \int \frac{x}{\sqrt{1 - x^{2}}} d x$

Let $1 - x^{2} = t$ , $- 2 x d x = d t \Rightarrow x d x = - \frac{1}{2} d t$ .

$\begin{array}{rcl} I_{1} & = & \int - \frac{1}{2 \sqrt{t}} d t \\ = & - \frac{1}{2} \int t^{- \frac{1}{2}} d t \\ = & - \frac{1}{2} \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} \\ = & - \frac{1}{2} \frac{t^{\frac{1}{2}}}{\frac{1}{2}} \\ = & - \frac{1}{2} 2 \sqrt{t} \\ = & - \sqrt{t} \\ = & - \sqrt{1 - x^{2}} \end{array}$

Step 4. Apply the limit of integration.

$\begin{array}{rcl} \int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{- 1} x & = & {[x \sin^{- 1} x + \sqrt{1 - x^{2}}]}_{- \frac{1}{2}}^{\frac{1}{2}} \\ = & [\frac{1}{2} \sin^{- 1} \frac{1}{2} + \sqrt{1 - {(\frac{1}{2})}^{2}}] - [- \frac{1}{2} \sin^{- 1} (- \frac{1}{2}) + \sqrt{1 - {(- \frac{1}{2})}^{2}}] \\ = & [\frac{1}{2} \sin^{- 1} \frac{1}{2} + \sqrt{1 - \frac{1}{4}}] - [- \frac{1}{2} \sin^{- 1} (- \frac{1}{2}) + \sqrt{1 - \frac{1}{4}}] \\ = & [\frac{1}{2} \sin^{- 1} \frac{1}{2} + \sqrt{\frac{3}{4}}] - [- \frac{1}{2} \sin^{- 1} (- \frac{1}{2}) + \sqrt{\frac{3}{4}}] \\ = & \frac{π}{12} + \frac{\sqrt{3}}{2} - \frac{π}{12} - \frac{\sqrt{3}}{2} \\ = & 0 \end{array}$

Step 5. Simplified answer

Hence, the required value is $0$ .

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Solve the definite integration.
$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{- 1} x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Solve for I1.

Step 4. Apply the limit of integration.

Step 5. Simplified answer

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Solve the definite integration.∫-1212sin-1xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Solve for I1.

Step 4. Apply the limit of integration.

Step 5. Simplified answer

One App. One Place for Learning.

Most popular questions from this chapter

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Solve the definite integration.
$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{- 1} x d x$