Indefinite Integrals

Front

What is an indefinite integral?

Back

An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed as ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration.

Front

What is the power rule for integration?

Back

The power rule states that ∫x^n dx = (1/(n+1))x^(n+1) + C, for n ≠ -1.

Front

Evaluate the indefinite integral: ∫ t⁶ dt

Back

1/7 t⁷ + C

Front

What does the constant of integration (C) represent in indefinite integrals?

Back

The constant of integration (C) represents the family of all antiderivatives of a function, accounting for the fact that the derivative of a constant is zero.

Front

Evaluate the indefinite integral: ∫ (3x² + 2) dx

Back

x³ + 2x + C

Front

What is the integral of a constant?

Back

The integral of a constant 'a' is ∫a dx = ax + C.

Front

Evaluate the indefinite integral: ∫ e^x dx

Back

e^x + C

Front

What is the relationship between differentiation and integration?

Back

Differentiation and integration are inverse operations; the derivative of an integral function returns the original function.

Front

Evaluate the indefinite integral: ∫ sin(x) dx

Back

-cos(x) + C

Front

What is the integral of x^n when n = -1?

Back

The integral of x^(-1) is ∫(1/x) dx = ln|x| + C.