The Logic Behind the Math: Fundamental Theorem

The core logic used in exams is the Fundamental Theorem of Calculus (FTC). It states that if $F(x)$ is the anti-derivative of $f(x)$, then:$$\int_a^b f(x) \, dx = F(b) - F(a)$$This allows us to find areas without doing complex summations; we simply find the "parent" function and subtract the values at the boundaries.

Step-by-Step Solved Example (Substitution Method)

Problem: Evaluate $\int 2x \cos(x^2) \, dx$.

• Step 1: Identify the "Inner" Function. Notice that $x^2$ is inside the cosine, and its derivative ($2x$) is also present. This is a clear case for u-substitution.
• Step 2: Assign $u$. Let $u = x^2$.
• Step 3: Differentiate $u$. $du/dx = 2x \implies du = 2x \, dx$.
• Step 4: Substitute into the Integral. The integral becomes $\int \cos(u) \, du$.
• Step 5: Integrate. The integral of $\cos(u)$ is $\sin(u) + C$.
• Step 6: Substitute Back. $\sin(x^2) + C$.

Alternative Methods: Integration by Parts

When two different types of functions are multiplied (e.g., $x \cdot e^x$), we use the formula $\int u \, dv = uv - \int v \, du$.The ILATE Rule: To choose $u$, follow the priority: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

Exam Trap Alert: The $+C$ and the Limits

In JEE Mains, students often lose marks on simple indefinite integrals by forgetting the constant $C$.

JEE Trap: In definite integrals, if you use substitution, you must change the limits to match the new variable $u$. Many students calculate $u$ but use the original $x$-limits, leading to an incorrect numerical value.

Practice Problem (JEE Main Level)

Question: Find the area bounded by the curve $y = x^2$ and the line $y = x$.Hint: Find the intersection points ($x=0, x=1$) and calculate $\int_0^1 (x - x^2) \, dx$.