Basic Integration and the FTC

Front

Define the Fundamental Theorem of Calculus (FTC).

Back

The Fundamental Theorem of Calculus states that if a function is continuous on the interval [a, b], then the integral of its derivative over that interval is equal to the difference in the values of the function at the endpoints: \( \int_a^b f'(x) \, dx = f(b) - f(a) \).

Front

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

Back

The integral of \( \sin x \) is \( -\cos x + C \), where C is the constant of integration.

Front

Calculate \( \int_{-2}^{-1} (4x^2 + 3x - 1) \, dx \).

Back

The result is \( \frac{23}{6} \).

Front

What is the integral of a polynomial function \( ax^n \)?

Back

The integral of a polynomial function \( ax^n \) is given by \( \frac{a}{n+1} x^{n+1} + C \) for \( n \neq -1 \).

Front

Evaluate \( \int_0^{\frac{\pi}{2}} \cos(x) \, dx \).

Back

The result is \( 1 \).

Front

What is the derivative of \( y = x^7 - x^3 + 5x + C \)?

Back

The derivative is \( \frac{dy}{dx} = 7x^6 - 3x^2 + 5 \).

Front

Calculate \( \int_1^5 (-x^2 + 6x - 10) \, dx \).

Back

The result is \( \frac{-28}{3} \).

Front

What is the integral of \( e^x \)?

Back

The integral of \( e^x \) is \( e^x + C \).

Front

Define the constant of integration (C).

Back

The constant of integration (C) represents an arbitrary constant added to the indefinite integral, accounting for all possible antiderivatives.

Front

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

Back

The integral of \( \cos x \) is \( \sin x + C \).