3.2 Q-2

Question Statement

Evaluate the following integrals:

i. $\int \frac{dx}{\sqrt{x+a} + \sqrt{x+b}} \quad {x+a > 0, x+b > 0}$

ii. $\int \frac{1 - x^2}{1 + x^2} , dx$

iii. $\int \frac{dx}{\sqrt{x+a} + \sqrt{x}} \quad {x > 0, a > 0}$

iv. $\int (a - 2x)^{\frac{3}{2}} , dx$

v. $\int \frac{(1 + e^x)^3}{e^x} , dx$

vi. $\int \sin((a+b)x) , dx$

vii. $\int \sqrt{1 - \cos(2x)} , dx \quad {1 - \cos(2x) > 0}$

viii. $\int (\ln x) \times \frac{1}{x} , dx$

ix. $\int \sin^2(x) , dx$ x. $\quad \int \frac{1}{1+\operatorname{Cos} x} \mathrm{dx}\left(\frac{\pi}{2}<x<\frac{\pi}{2}\right)$ xi. $\quad \int \frac{a x+b}{a x^{2}+2 b x+C} d x$

xii. $\int \cos 3 x \sin 2 x d x$ xiii. $\int \frac{\operatorname{Cos} 2 x-1}{1+\operatorname{Cos} 2 x} d x$

xiv. $\int \tan ^{2} x d x$

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Background and Explanation

To solve these integrals, you need knowledge of various integration techniques, including:

• Rationalizing integrals involving square roots.
• Using trigonometric identities and substitutions.
• Integration by substitution or direct application of basic formulas.
• Applying known integrals of logarithmic and trigonometric functions.

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Solution

i. $\int \frac{dx}{\sqrt{x+a} + \sqrt{x+b}}$

Step 1: Rationalize the denominator by multiplying by the conjugate:

$$\int \frac{dx}{\sqrt{x+a} + \sqrt{x+b}} \times \frac{\sqrt{x+a} - \sqrt{x+b}}{\sqrt{x+a} - \sqrt{x+b}}$$

This simplifies to:

$$\int \frac{(\sqrt{x+a} - \sqrt{x+b}) , dx}{(x+a) - (x+b)} = \frac{1}{a-b} \int (\sqrt{x+a} - \sqrt{x+b}) , dx$$

Step 2: Integrate each term:

$$\frac{1}{a-b} \left[ \frac{(x+a)^{\frac{3}{2}}}{\frac{3}{2}} - \frac{(x+b)^{\frac{3}{2}}}{\frac{3}{2}} \right] + C$$

Simplify to get the final answer:

$$\frac{2}{3(a-b)} \left[ (x+a)^{\frac{3}{2}} - (x+b)^{\frac{3}{2}} \right] + C$$

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ii. $\int \frac{1 - x^2}{1 + x^2} , dx$

Step 1: Break the fraction into two terms:

$$\int \left( \frac{2}{1 + x^2} - \frac{1 + x^2}{1 + x^2} \right) dx$$

Step 2: Integrate each term:

$$2 \tan^{-1}(x) - x + C$$

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iii. $\int \frac{dx}{\sqrt{x+a} + \sqrt{x}}$

Step 1: Rationalize the denominator by multiplying by the conjugate:

$$\int \frac{dx}{\sqrt{x+a} + \sqrt{x}} \times \frac{\sqrt{x+a} - \sqrt{x}}{\sqrt{x+a} - \sqrt{x}}$$

This simplifies to:

$$\frac{1}{a} \int (\sqrt{x+a} - \sqrt{x}) , dx$$

Step 2: Integrate each term:

$$\frac{1}{a} \left[ \frac{(x+a)^{\frac{3}{2}}}{\frac{3}{2}} - \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right] + C$$

Simplify to get the final answer:

$$\frac{1}{3a} \left[ (x+a)^{\frac{3}{2}} - x^{\frac{3}{2}} \right] + C$$

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iv. $\int (a - 2x)^{\frac{3}{2}} , dx$

Step 1: Multiply and divide by -2:

$$-\frac{1}{2} \int (a - 2x)^{\frac{3}{2}} (-2) dx$$

Step 2: Integrate using the power rule:

$$-\frac{1}{2} \cdot \frac{(a - 2x)^{\frac{5}{2}}}{\frac{5}{2}} + C = -\frac{1}{5} (a - 2x)^{\frac{5}{2}} + C$$

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v. $\int \frac{(1 + e^x)^3}{e^x} , dx$

Step 1: Expand the numerator and simplify:

$$\int \left( e^{-x} + 3 + 3 e^x + e^{2x} \right) dx$$

Step 2: Integrate each term:

$$-e^{-x} + 3x + 3e^x + \frac{e^{2x}}{2} + C$$

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vi. $\int \sin((a+b)x) , dx$

Step 1: Use substitution:

$$\frac{1}{a+b} \int \sin((a+b)x) (a+b) dx$$

Step 2: Integrate:

$$-\frac{1}{a+b} \cos((a+b)x) + C$$

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vii. $\int \sqrt{1 - \cos(2x)} , dx$

Step 1: Use the identity $1 - \cos(2x) = 2\sin^2(x)$:

$$\int \sqrt{2 \sin^2(x)} dx = \sqrt{2} \int \sin(x) dx$$

Step 2: Integrate:

$$-\sqrt{2} \cos(x) + C$$

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viii. $\int (\ln x) \times \frac{1}{x} , dx$

Step 1: Recognize that this integral is a standard form:

$$\int (\ln x) \times \frac{1}{x} , dx = \frac{(\ln x)^{2}}{2} + C$$

Step 2: Simplify the result:

$$\frac{(\ln x)^2}{2} + C$$

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ix. $\int \sin^2 x , dx$

Step 1: Use the identity for $\sin^2 x$:

$$\sin^2 x = \frac{1 - \cos 2x}{2}$$

Step 2: Substitute and simplify:

$$\int \sin^2 x , dx = \int \left( \frac{1 - \cos 2x}{2} \right) dx = \frac{1}{2} \int dx - \frac{1}{2} \int \cos 2x , dx$$

Step 3: Integrate each term:

$$\frac{1}{2} x - \frac{1}{2} \frac{\sin 2x}{2} + C$$

Step 4: Simplify the final expression:

$$\frac{1}{2} x - \frac{1}{4} \sin 2x + C$$

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x. $\int \frac{1}{1 + \cos x} , dx \quad \left( \frac{\pi}{2} < x < \frac{\pi}{2} \right)$

Step 1: Use the identity for $1 + \cos x$:

$$1 + \cos x = 2 \cos^2 \frac{x}{2}$$

Step 2: Substitute and simplify:

$$\int \frac{1}{1 + \cos x} , dx = \int \frac{1}{2 \cos^2 \frac{x}{2}} , dx = \frac{1}{2} \int \sec^2 \frac{x}{2} , dx$$

Step 3: Integrate using the standard formula for $\sec^2 x$:

$$\frac{1}{2} \tan \frac{x}{2} + C$$

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xi. $\int \frac{a x + b}{a x^2 + 2 b x + C} , dx$

Step 1: Multiply and divide by 2:

$$\int \frac{a x + b}{a x^2 + 2 b x + C} , dx = \frac{1}{2} \int \frac{2(a x + b)}{a x^2 + 2 b x + C} , dx$$

Step 2: Recognize the integral of the form $\frac{d}{dx} \ln |f(x)|$:

$$= \frac{1}{2} \ln |a x^2 + 2 b x + C| + C_1$$

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xii. $\int \cos 3x \sin 2x , dx$

Step 1: Use the identity for $2 \cos a x \sin b x$:

$$2 \cos a x \sin b x = \sin (a + b) + \sin (a - b)$$

Step 2: Simplify the integral:

$$\int \cos 3x \sin 2x , dx = \frac{1}{2} \int \left[ \sin(5x) + \sin(x) \right] , dx$$

Step 3: Integrate each term:

$$= \frac{1}{2} \left( \frac{-\cos 5x}{5} + \frac{-\cos x}{1} \right) + C$$

Step 4: Simplify:

$$= -\frac{1}{2} \left( \frac{\cos 5x}{5} + \cos x \right) + C$$

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xiii. $\int \frac{\cos 2x - 1}{1 + \cos 2x} , dx$

Step 1: Simplify using trigonometric identities:

$$\frac{\cos 2x - 1}{1 + \cos 2x} = - \frac{1 - \cos 2x}{1 + \cos 2x}$$

Step 2: Use the identity for $\cos 2x$:

$$= - \int \frac{2 \sin^2 x}{2 \cos^2 x} , dx = - \int \tan^2 x , dx$$

Step 3: Use the identity for $\tan^2 x$:

$$= - \int \left( \sec^2 x - 1 \right) , dx$$

Step 4: Integrate:

$$= - \tan x + x + C$$

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xiv. $\int \tan^2 x , dx$

Step 1: Use the identity for $\tan^2 x$:

$$\tan^2 x = \sec^2 x - 1$$

Step 2: Integrate:

$$\int \tan^2 x , dx = \int \sec^2 x , dx - \int 1 , dx$$

Step 3: Perform the integration:

$$= \tan x - x + C$$

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Key Formulas or Methods Used

1. Rationalization: Used to simplify integrals involving square roots in the denominator.
2. Power Rule: Used for integrating powers of x.
3. Trigonometric Substitutions: Used in integrals involving trigonometric identities.
4. Logarithmic Integration: Used to solve integrals involving logarithmic functions.

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Summary of Steps

1. Rationalize the denominator in integrals involving square roots.
2. Break down complex fractions into simpler terms.
3. Integrate each term separately.
4. Use substitution where needed for trigonometric or logarithmic integrals.
5. Simplify and apply the power rule where applicable.
6. Combine terms and add the constant of integration, (C).