Question Statement

Evaluate the indefinite integral:

$\int x e^{x^{2}} d x$

Background and Explanation

This integral requires the substitution method (or recognition of the chain rule in reverse). The key insight is that the derivative of $x^{2}$ is $2 x$ , which is proportional to the $x$ term multiplying the exponential.

Solution

Method 1: Recognition (Adjusting the Coefficient)

Let $I = \int x e^{x^{2}} d x$ .

We can rewrite the integral to match the form $\int e^{u} d u$ by adjusting for the derivative of the exponent:

$I = \int e^{x^{2}} \cdot x d x = \frac{1}{2} \int e^{x^{2}} \cdot (2 x) d x (multiplying and dividing by 2) = \frac{1}{2} e^{x^{2}} + c$

Here, we recognize that $(2 x) d x = d (x^{2})$ , so the integral becomes $\frac{1}{2} \int e^{x^{2}} d (x^{2}) = \frac{1}{2} e^{x^{2}} + c$ .

Method 2: Explicit Substitution (Alternative)

Alternatively, using explicit substitution as suggested by the fragments in the raw data:

Put $x^{2} = t$ (where the raw notation $x^{i}$ likely indicates $x^{2}$ ).

Differentiating both sides: $2 x d x = d t \Rightarrow x d x = \frac{1}{2} d t$

Substituting into the original integral: $\begin{aligned} I &= \int e^{t} \cdot \frac{1}{2} \, dt \\ &= \frac{1}{2} \int e^{t} \, dt \\ &= \frac{1}{2} e^{t} + c \\ &= \frac{1}{2} e^{x^{2}} + c \quad \text{(substituting back$ t = x^2 $)} \end{aligned}$

Key Formulas or Methods Used

Substitution Rule: $\int f (g (x)) \cdot g^{'} (x) d x = \int f (u) d u$ where $u = g (x)$
Exponential Integration: $\int e^{u} d u = e^{u} + C$
Chain Rule Recognition: $\int f^{'} (x) e^{f (x)} d x = e^{f (x)} + C$

Summary of Steps

Method 1 (Quick Recognition):

Identify that $x$ is proportional to the derivative of $x^{2}$ (since $\frac{d}{d x} (x^{2}) = 2 x$ ).
Multiply and divide by 2 to create the exact derivative $2 x$ inside the integral.
Integrate $e^{x^{2}}$ with respect to $x^{2}$ , yielding $\frac{1}{2} e^{x^{2}} + c$ .

Method 2 (Explicit Substitution):

Let $u = x^{2}$ (or $t = x^{2}$ ).
Compute $d u = 2 x d x$ , so $x d x = \frac{1}{2} d u$ .
Substitute to get $\frac{1}{2} \int e^{u} d u$ .
Integrate to get $\frac{1}{2} e^{u} + c$ , then substitute back $u = x^{2}$ .

Method 2: Explicit Substitution (Alternative)

Alternatively, using explicit substitution as suggested by the fragments in the raw data:

Put

x^{2} = t

(where the raw notation

x^{i}

likely indicates

x^{2}

Differentiating both sides:

2 x d x = d t \Rightarrow x d x = \frac{1}{2} d t

Substituting into the original integral:

\begin{aligned} I &= \int e^{t} \cdot \frac{1}{2} \, dt \\ &= \frac{1}{2} \int e^{t} \, dt \\ &= \frac{1}{2} e^{t} + c \\ &= \frac{1}{2} e^{x^{2}} + c \quad \text{(substituting back

t = x^2

)} \end{aligned}

Summary of Steps

Method 1 (Quick Recognition):

Identify that

x

is proportional to the derivative of

x^{2}

(since

\frac{d}{d x} (x^{2}) = 2 x

Multiply and divide by 2 to create the exact derivative

2 x

inside the integral.

Integrate

e^{x^{2}}

with respect to

x^{2}

, yielding

\frac{1}{2} e^{x^{2}} + c

Method 2 (Explicit Substitution):

Let

u = x^{2}

(or

t = x^{2}

Compute

d u = 2 x d x

, so

x d x = \frac{1}{2} d u

Substitute to get

\frac{1}{2} \int e^{u} d u

Integrate to get

\frac{1}{2} e^{u} + c

, then substitute back

u = x^{2}

3.1 Q-9

Question Statement

Background and Explanation

Solution

Method 1: Recognition (Adjusting the Coefficient)

Method 2: Explicit Substitution (Alternative)

Key Formulas or Methods Used

Summary of Steps

3.1 Q-9

Question Statement

Background and Explanation

Solution

Method 1: Recognition (Adjusting the Coefficient)

Method 2: Explicit Substitution (Alternative)

Key Formulas or Methods Used

Summary of Steps