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Worked problem · Calculus I · Units 1–3A

Fundamental Theorem Derivative: Worked Example and Solution

A complete course synthesis example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeCumulative progression

Problem

  1. Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dt.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

01

Problem 1: Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dt.

Answer: cos⁡(x2)\cos(x^2)

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

  1. FTC Part I evaluates the integrand at the variable upper bound: F′(x)=cos⁡(x2)F'(x)=\cos(x^2).
  2. The verified result is cos⁡(x2)\cos(x^2).

What is included

See a concise answer and full derivation for differentiate f (x) equals integral from 1 to x cosine (t squared) d t.

Skills assessed

  • Course synthesis

Prerequisites

  • Calculus I Units 1 through 3A

Common errors

  • Selecting a familiar formula without checking its hypotheses.

Learn fully

Supporting steps