Q14. Determine the value integral (1)/(^n(x)* ^n(x)) dx

Right 🌹CONFIDENTIALITY, let's continue.

Q14. Determine the value $\int \frac{1}{\sin^2(x) \cos^2(x)} \, dx$

Step 1: Use the identity $1 = \sin^2 x + \cos^2 x$ in the numerator. $\int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} \, dx$ Step 2: Split the fraction into two terms. $\int \left(\frac{\sin^2 x}{\sin^2 x \cos^2 x} + \frac{\cos^2 x}{\sin^2 x \cos^2 x}\right) \, dx$ Step 3: Simplify each term. $\int \left(\frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}\right) \, dx$ Step 4: Rewrite using reciprocal identities $\sec^2 x = \frac{1}{\cos^2 x}$ and $\csc^2 x = \frac{1}{\sin^2 x}$. $\int (\sec^2 x + \csc^2 x) \, dx$ Step 5: Integrate each term. $\int \sec^2 x \, dx = \tan x$ $\int \csc^2 x \, dx = -\cot x$ Step 6: Combine the results and add the constant of integration. $\tan x - \cot x + C$ The final answer is $\boxed{\tan x - \cot x + C}$.

Q15. Compute the integral: $\int \frac{1}{1+\sin x} \, dx$

Step 1: Multiply the numerator and denominator by the conjugate of the denominator, $(1-\sin x)$. $\int \frac{1}{1+\sin x} \cdot \frac{1-\sin x}{1-\sin x} \, dx$ Step 2: Simplify the denominator using the identity $1-\sin^2 x = \cos^2 x$. $\int \frac{1-\sin x}{1-\sin^2 x} \, dx = \int \frac{1-\sin x}{\cos^2 x} \, dx$ Step 3: Split the fraction into two terms. $\int \left(\frac{1}{\cos^2 x} - \frac{\sin x}{\cos^2 x}\right) \, dx$ Step 4: Rewrite the terms using $\sec^2 x = \frac{1}{\cos^2 x}$ and $\frac{\sin x}{\cos^2 x} = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x} = \tan x \sec x$. $\int (\sec^2 x - \sec x \tan x) \, dx$ Step 5: Integrate each term. $\int \sec^2 x \, dx = \tan x$ $\int \sec x \tan x \, dx = \sec x$ Step 6: Combine the results and add the constant of integration. $\tan x - \sec x + C$ The final answer is $\boxed{\tan x - \sec x + C}$.

PART II: OBJECTIVE QUESTIONS (MULTIPLE CHOICE)

1. Determine $\int 4dx$.

Step 1: Apply the constant rule for integration, $\int k \, dx = kx + C$. $\int 4 \, dx = 4x + C$ Step 2: Compare with the given options. The correct option is (B). The final answer is $\boxed{B. 4x+C}$.

2. Find $\int x^{\frac{1}{2}} dx$.

Step 1: Apply the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. $\int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C$ Step 2: Simplify the exponent and denominator. $\frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3}x^{\frac{3}{2}} + C$ Step 3: Compare with the given options. The correct option is (A). The final answer is $\boxed{A. \frac{2}{3}x^{\frac{3}{2}}+C}$.

3. Determine $\int_1^2 5x^2 dx$.

Step 1: Find the indefinite integral of $5x^2$. $\int 5x^2 \, dx = 5 \cdot \frac{x^{2+1}}{2+1} + C = 5 \cdot \frac{x^3}{3} + C = \frac{5}{3}x^3 + C$ Step 2: Evaluate the definite integral using the Fundamental Theorem of Calculus, $F(b) - F(a)$. $\left[\frac{5}{3}x^3\right]_1^2 = \left(\frac{5}{3}(2)^3\right) - \left(\frac{5}{3}(1)^3\right)$ Step 3: Calculate the numerical value. $\frac{5}{3}(8) - \frac{5}{3}(1) = \frac{40}{3} - \frac{5}{3} = \frac{35}{3}$ Step 4: Convert to a decimal or mixed number to compare with options. $\frac{35}{3} \approx 11.67$ Looking at the options (A) 305, (B) 95, (C) 105, (D) 205, it seems there might be a typo in the question or the options provided, as $\frac{35}{3}$ is not among them. Assuming the question intended to ask for the indefinite integral or there's a different problem, I've provided the correct calculation for the given definite integral. If the question intended to ask for something else, please clarify.

Based on the calculation, none of the options (A) 305, (B) 95, (C) 105, (D) 205 match $\frac{35}{3}$.

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