Q. 42 Combining derivatives and integr... [FREE SOLUTION]

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Chapter 4: Q. 42 (page 405)

Combining derivatives and integrals: Simplify each of the following as much as possible:
${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x$

Short Answer

Expert verified

The solution is ${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = [\frac{|x^{2} + 1|}{5}] .$

Step by step solution

Step 1. Given information

Simplify: ${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x$

Step 2. Calculation

The given equation is:

${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x$

$\frac{d}{d x} \ln x = \frac{1}{x}$ & Chain rule

${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = {鈭�}_{- 2}^{x} (\frac{1}{(x^{2} + 1) d x} \frac{d}{d x} (x^{2} + 1)) d x {鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = {鈭�}_{- 2}^{x} (\frac{2 x}{(x^{2} + 1)}) d x L e t (x^{2} + 1) = t$

Differentiate both sides with respect to 't'

$\frac{d}{d t} (x^{2} + 1) = \frac{d t}{d t} 2 x d x = d t$

Also when

$x = - 2 t h e n t = 5 & w h e n x = x t h e n t = (x^{2} + 1)$

${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = {鈭�}_{5}^{x^{2} + 1} (\frac{1}{t}) d t U \sin g 鈭� \frac{1}{x} d x = \ln |x| + C {鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = {[\ln |t|]}_{5}^{x^{2} + 1} {鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = [\ln |x^{2} + 1| - \ln 5] {鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x = [\ln \frac{|x^{2} + 1|}{5}]$

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91影视

Combining derivatives and integrals: Simplify each of the following as much as possible:
${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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Combining derivatives and integrals: Simplify each of the following as much as possible:鈭�-2xddx(ln(x2+1))dx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Study anywhere. Anytime. Across all devices.

Combining derivatives and integrals: Simplify each of the following as much as possible:
${鈭�}_{- 2}^{x} \frac{d}{d x} (\ln (x^{2} + 1)) d x$