Question: Assignment Activity Unit 7 - Due Aug 5, 2026 11:55 PM
• MATH 1211-01 Calculus - AY2026-T5
This assignment will assess your knowledge and skills in:
• Explaining the concept of antiderivatives and demonstrating how to compute indefinite integrals.
• Performing basic integration techniques, including substitution, to solve integrals.
Assignment
Please answer the following questions and explain your reasoning wherever necessary.
Question 1: Find the antiderivative of each function, providing all necessary steps.
I. f(x) = 1/4 - 2/5 x^3
II. f(x) = 3cosx - 4sec^2 x
III. f(x) = 2√x - 3/(2√(x))
IV. f(x) = cosxe^(sinx)
Question 2: Evaluate the following integrals with all the required steps.
I. ∫(5e^2x − 1/(x√x))dx
II. ∫(sin2x − 2cosx)dx
III. ∫3secxtanxdx
IV. ∫(3/x − 2/(2x−1))dx
Question 3: Use appropriate substitution to evaluate the following integrals with all the required steps.
I. ∫(5sec^2x/(3+tanx))dx
II. ∫(a^3x^3/√(a^2−a^4x^4))dx where a is constant
III. ∫cosx sin^4x dx
IV. ∫(4x^3 + 2x)e^(x^4+x^2) dx
1. **Problem:** Find the antiderivative of $f(x) = \frac{1}{4} - \frac{2}{5}x^3$.
2. **Formula:** The antiderivative (indefinite integral) of $x^n$ is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
3. **Step-by-step:**
$$\int \left(\frac{1}{4} - \frac{2}{5}x^3\right) dx = \int \frac{1}{4} dx - \int \frac{2}{5}x^3 dx$$
$$= \frac{1}{4}x - \frac{2}{5} \cdot \frac{x^{4}}{4} + C = \frac{1}{4}x - \frac{2}{5} \cdot \frac{x^{4}}{4} + C$$
$$= \frac{1}{4}x - \frac{2}{20}x^{4} + C = \frac{1}{4}x - \frac{1}{10}x^{4} + C$$
4. **Final answer:**
$$F(x) = \frac{1}{4}x - \frac{1}{10}x^{4} + C$$