Integral Calculus

Front

What is the integral of a constant function?

Back

The integral of a constant function \( c \) is given by \( cx + C \), where \( C \) is the constant of integration.

Front

Find \( f(x) \) if \( f'(x) = 3x^2 \).

Back

\( f(x) = x^3 + C \)

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus states that if \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \), then \( \int_a^b f(x)dx = F(b) - F(a) \).

Front

Evaluate \( \int x^3 dx \).

Back

\( \frac{1}{4}x^4 + C \)

Front

What is an antiderivative?

Back

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \).

Front

Evaluate \( \int 5x^4 dx \).

Back

\( x^5 + C \)

Front

What is the relationship between differentiation and integration?

Back

Differentiation and integration are inverse processes; integration undoes differentiation.

Front

Find \( \int e^{3x} dx \).

Back

\( \frac{1}{3}e^{3x} + C \)

Front

What does the constant of integration represent?

Back

The constant of integration \( C \) represents an infinite number of possible vertical shifts of the antiderivative.

Front

Evaluate \( \int (2x + 3) dx \).

Back

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