Formula: d/dx [integral(g(x) to h(x)) f(t) dt] = f(h(x)) * h'(x) - f(g(x)) * g'(x)

Special Cases:

• Fixed lower limit: d/dx [integral(a to h(x)) f(t) dt] = f(h(x)) * h'(x)
• d/dx [integral(0 to x) f(t) dt] = f(x) (FTC-1)
• d/dx [integral(0 to $x^2$) f(t) dt] = f($x^2$) * 2x

JEE Pattern: Given integral(0 to x) f(t) dt = some expression in x, differentiate both sides to find f(x).

Example: If integral(0 to x) f(t) dt = x + integral(x to 1) tf(t) dt, differentiate: f(x) = 1 - xf(x). So f(x) = $\frac{1}{1+x}$.