I can solve this for you! The problem requires evaluating definite integrals $I$ for the given expressions. Let's solve each part step by step.

a) $I = \int_0^1 x e^{x-1} dx$

We will use substitution or other integration techniques to compute this.

b) $I = \int_0^1 \frac{2x-1}{x+1} dx$

This involves simplifying the fraction or using substitution for the logarithmic function.

c) $I = \int_0^1 \frac{x^2 - x + 1}{x+1} dx$

Simplify the numerator relative to the denominator to split into simpler terms.

d) $I = \int_0^1 x \sqrt{x^2 + 1} dx$

This will likely require substitution such as $u = x^2 + 1$.

f) $I = \int_0^1 \frac{x+3}{x^2 + 2x + 1} dx$

The denominator can be rewritten as $(x+1)^2$, simplifying the expression.

Let me calculate each integral step-by-step and provide formulas or tips as I go. Stay tuned! sympy as sp

Define the variable

x = sp.symbols('x')

(a) I = ∫ x * exp(x-1) dx from 0 to 1

I_a = sp.integrate(x * sp.exp(x - 1), (x, 0, 1))

(b) I = ∫ (2x - 1) / (x + 1) dx from 0 to 1

I_b = sp.integrate((2 * x - 1) / (x + 1), (x, 0, 1))

(c) I = ∫ (x^2 - x + 1) / (x + 1) dx from 0 to 1

I_c = sp.integrate((x**2 - x + 1) / (x + 1), (x, 0, 1))

(d) I = ∫ x * sqrt(x^2 + 1) dx from 0 to 1

I_d = sp.integrate(x * sp.sqrt(x**2 + 1), (x, 0, 1))

(f) I = ∫ (x + 3) / (x^2 + 2x + 1) dx from 0 to 1

I_f = sp.integrate((x + 3) / (x**2 + 2 * x + 1), (x, 0, 1))

I_a, I_b, I_c, I_d, I_fHere are the solutions to each definite integral:

Solutions:

1. (a) $I = \int_0^1 x e^{x-1} dx = e^{-1}$
2. (b) $I = \int_0^1 \frac{2x-1}{x+1} dx = 2 - 3 \ln(2)$
3. (c) $I = \int_0^1 \frac{x^2 - x + 1}{x+1} dx = -\frac{3}{2} + 3 \ln(2)$
4. (d) $I = \int_0^1 x \sqrt{x^2 + 1} dx = -\frac{1}{3} + \frac{2 \sqrt{2}}{3}$
5. (f) $I = \int_0^1 \frac{x+3}{x^2 + 2x + 1} dx = \ln(2) + 1$

Explanation of Methods:

• For (a), direct integration was used involving exponential functions.
• For (b) and (c), partial fractions or logarithmic simplifications helped solve the integrals.
• For (d), substitution $u = x^2 + 1$ simplifies the square root.
• For (f), rewriting $x^2 + 2x + 1$ as $(x+1)^2$ makes the integral easier.

Relevant Formulas:

1. Exponential Integrals: $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$
2. Logarithmic Derivatives: $\int \frac{1}{x} dx = \ln|x| + C$
3. Substitution Method: For $u = g(x)$, $\int f(g(x)) g'(x) dx = \int f(u) du$
4. Integration by Parts: $\int u dv = uv - \int v du$
5. Rational Functions: Decompose into partial fractions if the numerator degree < denominator degree.

Would you like detailed steps for any of these integrals?

Related Questions:

1. Can you show how substitution simplifies the integral in (d)?
2. How does logarithmic simplification work in (b)?
3. Can we verify (f) using partial fractions instead?
4. Are there any general rules for integrals of the form $\int \frac{P(x)}{Q(x)} dx$?
5. Why is exponential decay important in (a)?

Tip: Always simplify the integrand if possible before applying integration techniques. It reduces computational errors and speeds up solving!