7–84. Evaluate the following integrals.
35. ∫ from 0 to π/4 [(tan²θ + tanθ + 1) sec²θ] dθ

Verified step by step guidance

1. Step 1: Recognize that the integral involves the expression (tan²θ + tanθ + 1) multiplied by sec²θ. Start by analyzing the structure of the integrand to see if simplifications or substitutions can be applied.
2. Step 2: Recall that the derivative of tan(θ) is sec²(θ). This suggests that a substitution u = tan(θ) might simplify the integral. Compute du = sec²(θ) dθ.
3. Step 3: Substitute u = tan(θ) into the integral. The limits of integration will change accordingly: when θ = 0, u = tan(0) = 0; and when θ = π/4, u = tan(π/4) = 1. The integral becomes ∫ from 0 to 1 [(u² + u + 1)] du.
4. Step 4: Break the integral into simpler parts: ∫ from 0 to 1 u² du + ∫ from 0 to 1 u du + ∫ from 0 to 1 1 du. Compute each term separately using the power rule for integration: ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C.
5. Step 5: After integrating each term, combine the results and evaluate the definite integral by substituting the limits of integration (0 and 1). This will yield the final value of the integral.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and is used to calculate quantities such as areas, volumes, and total accumulated change. In this problem, we are tasked with evaluating a definite integral, which requires applying the appropriate integration techniques to find the exact value over the specified limits.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this integral, recognizing and applying identities such as the Pythagorean identity or the relationships between tangent and secant can simplify the expression. Understanding these identities is crucial for transforming the integrand into a more manageable form for integration.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This method is particularly useful when the integrand contains composite functions. In this case, substituting a trigonometric function can help to rewrite the integral in a simpler form, making it easier to evaluate the integral over the given limits.