Integrate the following function: $ \frac{(x + 1)(x + \log x)^2}{x} $

Integrate the following function: ((x + 1)(x + log x)²)/(x)

Exercise 7.2 · Q36

\frac{(x + 1)(x + \log x)^2}{x}

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Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The chain rule differentiates composite functions: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ — differentiate the outer function, then multiply by the derivative of the inside. Integration asks the reverse: given $\cos(x^2) \cdot 2x$ , find the original function. That's what u substitution does — it reverses the chain rule.

The Core Intuition

When an integral looks like "a function times the derivative of its inside," substitute the inside with $u$ and the derivative of the inside with $du$ . Consider:

\int 2x \cos(x^2) \, dx

Here $2x$ is the derivative of $x^2$ , and $x^2$ is the inside of $\cos(x^2)$ . Let $u = x^2$ , so $du = 2x \, dx$ :

\int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C

Check: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ .

The Precise Statement

\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x), \, du = g'(x) \, dx

Valid provided $g$ is differentiable and the resulting integral in $u$ is simpler.

The Step-by-Step Method

Identify a function $g(x)$ whose derivative $g'(x)$ also appears (possibly up to a constant factor).
Set $u = g(x)$ , compute $du = g'(x) \, dx$ .
Rewrite the entire integral in $u$ and $du$ — every $x$ and $dx$ must be replaced.
Integrate with respect to $u$ .
Substitute back $u = g(x)$ .

Watch out

You cannot mix variables. If any $x$ remains after substitution, you chose the wrong $u$ (or must solve for $x$ in terms of $u$ — rare).

A Second Example (with a constant factor)

Evaluate $\int x \sqrt{x^2 + 1} \, dx$ . Let $u = x^2 + 1$ , so $x \, dx = \frac{1}{2} du$ :

\int \sqrt{u} \cdot \tfrac{1}{2} du = \tfrac{1}{2} \cdot \tfrac{2}{3} u^{3/2} + C = \tfrac{1}{3} (x^2 + 1)^{3/2} + C

When Does It Work?

When the integrand is something times the derivative of something inside. Common patterns:

$x \cdot f(x^2)$ — derivative of $x^2$ is $2x$ , so $u = x^2$
$e^{g(x)} \cdot g'(x)$ — derivative of $g(x)$ appears
$\frac{g'(x)}{g(x)}$ — leads to $\log|g(x)|$ …

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The key idea is to rewrite the integrand so that the derivative of $(x + \log x)$ appears as a factor, enabling a direct $u$ -substitution. The integral evaluates to $\frac{(x + \log x)^3}{3} + C$ .

Why substitution works here

When you see an expression like $(x + \log x)^2$ multiplied by something, your first instinct should be: can I find the derivative of the inside function somewhere in the integrand? Here, the inside function is $u = x + \log x$ , and its derivative is $1 + \frac{1}{x}$ . The given integrand is $\frac{(x+1)(x + \log x)^2}{x}$ . Notice that $\frac{x+1}{x} = 1 + \frac{1}{x}$ , which is exactly $u'$ . That’s the signal — the whole integrand is $u' \cdot u^2$ , a perfect candidate for the reverse chain rule.

Tip

Spotting $u'$ multiplied by a power of $u$ is the most common pattern for $u$ -substitution. If you see $(something)^n$ and its derivative nearby, substitution is almost always the path.

Step-by-step solution

1. Rewrite the integrand to reveal the pattern

Break the fraction into two simpler terms:

\frac{(x + 1)(x + \log x)^2}{x} = \left( \frac{x+1}{x} \right) (x + \log x)^2 = \left(1 + \frac{1}{x}\right) (x + \log x)^2

Now it’s clear: the factor $1 + \frac{1}{x}$ is the derivative of $x + \log x$ .

2. Perform the substitution

Let $u = x + \log x$ . Then: …

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Q.Integrate the following function: $\frac{(x + 1)(x + \log x)^2}{x}$

Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The Core Intuition

The Precise Statement

The Step-by-Step Method

A Second Example (with a constant factor)

When Does It Work?

Why substitution works here

Step-by-step solution

Method: Recognise the $g'(x)\,[g(x)]^n$ pattern and substitute the inner function

Steps

Common Mistakes

Q.Integrate the following function: (x+1)(x+log⁡x)2x\frac{(x + 1)(x + \log x)^2}{x}x(x+1)(x+logx)2​

Why substitution works here

Step-by-step solution

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Q.Integrate the following function: $\frac{(x + 1)(x + \log x)^2}{x}$

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