The value of integral34 2x/1+x2 dx is

Option b: log 17/10. Solution: We need to evaluate the definite integral: I = integral34 2x/1+x2 dx Let u = 1 + x2 . Differentiating both sides: du = 2x dx Change the integration limits for u : - When x = 3 : u = 1 + 32 = 10 - When x = 4 : u = 1 + 42 = 17 Substitute these into the integral: I = integral1017 1/u du I = [ log(u) ]1017 I = log(17) - log(10) = log(17/10) Hence, Option B is the correct answer.

The value of $\int_{3}^{4} \frac{2x}{1+x^2} dx$ is

The correct answer is Option b: \log \frac{17}{10}. We need to evaluate the definite integral: $$I = \int_{3}^{4} \frac{2x}{1+x^2} dx$$ Let $u = 1 + x^2$. Differentiating both sides: $$du = 2x \, dx$$ Change the integration limits for $u$: - When $x = 3$: $u = 1 + 3^2 = 10$ - When $x = 4$: $u = 1 + 4^2 = 17$ Substitute these into the integral: $$I = \int_{10}^{17} \frac{1}{u} du$$ $$I = \left[ \log(u) \right]_{10}^{17}$$ $$I = \log(17) - \log(10) = \log\left(\frac{17}{10}\right)$$ Hence, **Option B** is the correct answer.

Basic Applications of CalculusPYQ Sept 25Question 4139 of 28

All Questions

The value of $\displaystyle \int_{3}^{4} \frac{2x}{1+x^2} dx$ is

Options

\displaystyle \log \frac{16}{10}

\displaystyle \log \frac{17}{10}

\displaystyle \log \frac{16}{9}

\displaystyle \log \frac{17}{9}

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Correct Answer

✅ Option b — $\displaystyle \log \frac{17}{10}$

All Options:

A $\displaystyle \log \frac{16}{10}$
B $\displaystyle \log \frac{17}{10}$
C $\displaystyle \log \frac{16}{9}$
D $\displaystyle \log \frac{17}{9}$

Detailed Solution & Explanation

We need to evaluate the definite integral:

I = \int_{3}^{4} \frac{2x}{1+x^2} dx

Let

\displaystyle u = 1 + x^2

. Differentiating both sides:

du = 2x \, dx

Change the integration limits for

\displaystyle u

: - When

\displaystyle x = 3

\displaystyle u = 1 + 3^2 = 10

- When

\displaystyle x = 4

\displaystyle u = 1 + 4^2 = 17

Substitute these into the integral:

I = \int_{10}^{17} \frac{1}{u} du

I = \left[ \log(u) \right]_{10}^{17}

I = \log(17) - \log(10) = \log\left(\frac{17}{10}\right)

Hence, **Option B** is the correct answer.

About This Chapter: Basic Applications of Calculus

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Limits, Continuity, Derivatives, Integrals

This chapter covers Limits, Continuity, Derivatives, Integrals and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

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The value of ∫342x1+x2dx\displaystyle \int_{3}^{4} \frac{2x}{1+x^2} dx∫34​1+x22x​dx is