Chapter 5: Q 90. (page 431)

Prove the integration formula.
$\int \sin^{- 1} x d x = x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$
(a) by applying integration by parts to $\int \sin^{- 1} x d x$ .
(b) by differentiating $\sqrt{1 - x^{2}} + x \sin^{- 1} x$ .

Short Answer

Expert verified

Part (a). The solution is $x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$ .

Part (b). The solution is $\sin^{- 1} x$ .

Step by step solution

Part (a) Step 1. Given information.

The given formula is:

$\int \sin^{- 1} x d x = x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$ .

Part (a) Step 2. Prove the formula by applying the integration by parts formula.

Take $u = \sin^{- 1} x$ and $v = 1$ .

role="math" localid="1650366363299" $\begin{array}{rcl} \int \sin^{- 1} x \cdot 1 d x & = & \sin^{- 1} x \int d x - \int [\frac{d}{d x} (\sin^{- 1} x) \int d x] d x \\ = & \sin^{- 1} x \cdot x - \int \frac{1}{\sqrt{1 - x^{2}}} \cdot x d x \\ = & x \sin^{- 1} x - \int \frac{1}{\sqrt{t}} (\frac{- d t}{2}) \dots \dots \dots \dots \{p u t 1 - x^{2} = t, - 2 x d x = d t, x d x = - \frac{d t}{2}\} \\ = & x \sin^{- 1} x + \frac{1}{2} \int \frac{1}{\sqrt{t}} d t \\ = & x \sin^{- 1} x + \frac{1}{2} \int t^{- \frac{1}{2}} d t \\ = & x \sin^{- 1} x + \frac{1}{2} \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} \\ = & x \sin^{- 1} x + \frac{1}{2} \cdot 2 \cdot \sqrt{t} + C \\ = & x \sin^{- 1} x + \sqrt{t} + C \end{array}$

Substitute $t = \sqrt{1 - x^{2}}$ , in the above result.

Hence, the formula $x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$ is proved.

Part (b) Step 1. Given information.

The given formula is:

$\int \sin^{- 1} x d x = x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$

Part (b) Step 2. Prove the formula by differentiating x·sin-1x+1-x2.

The differentiation is calculated below:

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

Hence, the formula is proved.

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Prove the integration formula.
$\int \sin^{- 1} x d x = x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$
(a) by applying integration by parts to $\int \sin^{- 1} x d x$ .
(b) by differentiating $\sqrt{1 - x^{2}} + x \sin^{- 1} x$ .

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Prove the formula by applying the integration by parts formula.

Part (b) Step 1. Given information.

Part (b) Step 2. Prove the formula by differentiating x·sin-1x+1-x2.

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Prove the integration formula.∫sin-1xdx=xsin-1x+1-x2+C(a) by applying integration by parts to ∫sin-1xdx.(b) by differentiating1-x2+xsin-1x.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Prove the formula by applying the integration by parts formula.

Part (b) Step 1. Given information.

Part (b) Step 2. Prove the formula by differentiating x·sin-1x+1-x2.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Prove the integration formula.
$\int \sin^{- 1} x d x = x \sin^{- 1} x + \sqrt{1 - x^{2}} + C$
(a) by applying integration by parts to $\int \sin^{- 1} x d x$ .
(b) by differentiating $\sqrt{1 - x^{2}} + x \sin^{- 1} x$ .