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 \( \int f(x) \, dx = F(x) + C \), 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 for any real number \( n \neq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

Front

Evaluate the indefinite integral: \( \int 4 \, dx \)

Back

\( 4x + C \)

Front

Evaluate the indefinite integral: \( \int (16x^3 + 5 - 6x^{-3}) \, dx \)

Back

\( 4x^4 + 5x + 3x^{-2} + C \)

Front

What is the antiderivative of \( \sqrt{x} \)?

Back

\( \int \sqrt{x} \, dx = \frac{2}{3}x^{rac{3}{2}} + C \)

Front

What does the constant of integration represent in indefinite integrals?

Back

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

Front

Evaluate the indefinite integral: \( \int (3x^2 + 2x + 1) \, dx \)

Back

\( x^3 + x^2 + x + C \)

Front

What is the integral of a constant \( k \)?

Back

\( \int k \, dx = kx + C \) where \( k \) is a constant.

Front

Evaluate the indefinite integral: \( \int e^x \, dx \)

Back

\( e^x + C \)

Front

What is the integral of \( \sin(x) \)?

Back

\( \int \sin(x) \, dx = -\cos(x) + C \)