Chapter 5: Q. 6 (page 494)

If $f$ and $u$ are functions such that role="math" localid="1652080830652" $f' (u (x)) (u' (x))$ is role="math" localid="1652080865441" , then we can use integration by substitution to solve
role="math" localid="1652080904444" $\int f' (u (x)) (u' (x)) d x =$

Short Answer

Expert verified

$\int f' (u (x)) (u' (x)) d x = f (u (x)) + C$

Step by step solution

Step 1.

We know that in any integral $\int f (x) d x$ , $f (x)$ is integrand.

And If localid="1652080706482" $\int f' (u (x)) (u' (x)) d x$ is needed to be integrated by substitution then

We substitute localid="1652080668665" $u (x) = v$

hence,localid="1652080598731" $\frac{d v}{d x} = u' (x) d v = u' (x) . d x$

Thus the given integral become

localid="1652080932120" $\int f' (u (x)) (u' (x)) d x = \int f' (v) d v \Rightarrow \int f' (u (x)) (u' (x)) d x = f (v) + C$

Substituting back

We get, $\int f' (u (x)) (u' (x)) d x = f (u (x)) + C$

Step 3. Final Answer

$f (u (x)) + C$ If $f$ and $u$ are functions such that $f' (u (x)) (u' (x))$ is $i n t e g r a n d$ , then we can use integration by substitution to solve

$\int f' (u (x)) (u' (x)) d x = f (u (x)) + C$

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If $f$ and $u$ are functions such that role="math" localid="1652080830652" $f' (u (x)) (u' (x))$ is role="math" localid="1652080865441" , then we can use integration by substitution to solve
role="math" localid="1652080904444" $\int f' (u (x)) (u' (x)) d x =$

Short Answer

Step by step solution

Step 1.

Step 3. Final Answer

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If fand uare functions such that role="math" localid="1652080830652" f'(u(x))u'(x)is role="math" localid="1652080865441" , then we can use integration by substitution to solverole="math" localid="1652080904444" ∫f'(u(x))u'(x)dx=

Short Answer

Step by step solution

Step 1.

Step 3. Final Answer

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

If $f$ and $u$ are functions such that role="math" localid="1652080830652" $f' (u (x)) (u' (x))$ is role="math" localid="1652080865441" , then we can use integration by substitution to solve
role="math" localid="1652080904444" $\int f' (u (x)) (u' (x)) d x =$