https://file.aiquickdraw.com/mathbot/file/17262383153372z9huwpr.pdf

The PDF contains three integral calculus problems, including integrals that require techniques such as substitution, logarithmic identities, trigonometric integrals, and partial fraction decomposition. Let's solve each one step by step.

### 1. Evaluate the following integrals:

#### (a) \(\int 2z^5 e^{1-z} dz\)

We can use **substitution** to solve this.

Let \( u = 1 - z \), so \( du = -dz \).

Now the integral becomes:
\[
-2 \int z^5 e^u du
\]
Next, substitute back \( z = 1 - u \), and apply integration techniques to solve.

---

#### (b) \(\int \left[ \ln(w^2) \right]^4 \frac{1}{w} dw\)

Use the logarithmic identity \( \ln(w^2) = 2\ln(w) \) to simplify the integrand. The integral becomes:
\[
\int (2\ln(w))^4 \frac{1}{w} dw = 16 \int \ln^4(w) \frac{1}{w} dw
\]
This can be solved using repeated integration by parts or a reduction formula for powers of logarithms.

---

### 2. Solve the integral \(\int \sin^7(x) dx\) by using trigonometric identities.

We can express \(\sin^7(x)\) as:
\[
\sin^7(x) = \sin^6(x) \sin(x) = (\sin^2(x))^3 \sin(x)
\]
Then, use the identity \( \sin^2(x) = 1 - \cos^2(x) \) to expand the expression and integrate using substitution.

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### 3. Evaluate the integral \(\int \frac{9 - w^2}{3w^3 + 10w^2 - 13w - 20} dw\) using partial fraction decomposition.

First, factor the denominator \( 3w^3 + 10w^2 - 13w - 20 \) to express the integrand in terms of simpler fractions. Then apply partial fraction decomposition to integrate each term separately.

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Would you like more details on any of the steps, or a specific solution broken down further?

Here are 5 related questions to explore:

1. How do you perform substitution in integrals with exponential functions?
2. What is the reduction formula for integrals involving powers of logarithms?
3. How can trigonometric identities simplify powers of sine or cosine in integrals?
4. How do you factor complex cubic polynomials for partial fractions?
5. What are common methods to approach integrals involving logarithmic terms?

**Tip:** For integrals involving powers of sine or cosine, converting them to terms of powers of \(\cos(x)\) or \(\sin(x)\) using Pythagorean identities simplifies the process.

Solve complex integral calculus problems involving exponential, logarithmic, and trigonometric functions, using substitution, logarithmic identities, and partial fraction decomposition. This is designed for undergraduate students studying integral calculus.

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