Verify that [math]\frac1{r+1} x^{r+1}[/math] is an antiderivative of [math]x^r[/math], if [math]r[/math] is any rational number except [math]-1[/math].

Find an antiderivative of each of the following functions.

• [math]f(x) = x^7[/math]
• [math]f(x) = x^3 + \frac1{x^3}[/math]
• [math]f(y) = 7y^\frac15[/math]
• [math]f(t) = 5t^4 + 3t^2 + 1[/math]
• [math]g(x) = (3x + 1)^2[/math]
• [math]f(x) = \frac{2x}{(x^2 + 1)^2}[/math].

Evaluate each of the following definite integrals by finding an antiderivative and using Theorem.

• [math]\int_0^1 3x^2 \; dx[/math]
• [math]\int_0^1 (4x^3 + 3x^2 + 2x + 1) \; dx[/math]
• [math]\int_1^3 (5x - 1) \; dx[/math]
• [math]\int_1^3 (5t - 1) \; dt[/math]
• [math]\int_1^2 \left( x^2 + \frac1{x^2} \right) \; dx[/math]
• [math]\int_3^2 x^\frac13 \; dx[/math]
• [math]\int_{-2}^0 y^\frac15 \; dy[/math]
• [math]\int_1^2 \left( \frac2{x^3} + \frac1{x^2} + 2 \right) \; dx[/math]
• [math]\int_{-1}^1 (y^2 - y + 1) \; dy[/math]
• [math]\int_6^0 (x^3 - 9x^2 + 16x) \; dx[/math]
• [math]\int_3^5 (2x - 1)^2 \; dx[/math]
• [math]\int_3^x (6t^2 - 4t + 2) \; dx[/math]
• [math]\int_0^t (x^2 + 3x - 1) \; dx[/math]
• [math]\int_0^{x^2} s^3 \; ds[/math]
• [math]\int_a^b dx[/math]
• [math]\int_x^{3x} (4t - 1) \; dt[/math].

Let [math]n[/math] be a positive integer.

• Evaluate [math]\int_a^b x^n \; dx[/math].
• lab{4.5.4b} Evaluate [math]\int_a^b \frac1{x^n} \; dx[/math] provided (i) [math]n \ne 1[/math], and (ii) [math]a[/math] and [math]b[/math] are either both positive or both negative.
• In \ref{ex4.5.4b}, what is the reason for proviso (i)? For proviso (ii)?

Let [math]F[/math] be an antiderivative of [math]f[/math] and [math]G[/math] an antiderivative of [math]g[/math].

• Prove that [math]F+G[/math] is an antiderivative of [math]f+g[/math].
• For any constant [math]k[/math], prove that [math]kF[/math] is an antiderivative of [math]kf[/math].

What is the domain of the function [math]F[/math] defined by [math]F(t) = \int_1^t \frac1x \; dx[/math].

Let [math]F(t) = \int_0^t (6x^2 - 4x + 1) \; dx[/math].

• lab{4.5.7a} Using just the Fundamental Theorem and without evaluating [math]F[/math], find [math]F^\prime(t)[/math], [math]F^\prime(-1)[/math], [math]F^\prime(2)[/math], and [math]F^\prime(x)[/math].
• lab{4.5.7b} Find [math]F(t)[/math] as a polynomial in [math]t[/math] by finding a polynomial which is an antiderivative of [math]6x^2 - 4x + 1[/math].
• Differentiate the answer in \ref{ex4.5.7b}, and thereby check \ref{ex4.5.7a}.

Let [math]G(x) = \int_1^x \left( t + \frac1{t^2} \right) \; dt[/math], for [math]x \gt 0[/math].

• lab{4.5.8a} Using just the Fundamental Theorem, find [math]G^\prime(x)[/math] and [math]G^\prime(2)[/math].
• lab{4.5.8b} Evaluate [math]G(x)[/math] as a rational function of [math]x[/math] by finding an antiderivative of [math]t + \frac1{t^2}[/math].
• Take the derivative of [math]G(x)[/math] as found in \ref{ex4.5.8b} and thereby check \ref{ex4.5.8a}.

• lab{4.5.9a} Evaluate [math]F(t) = \int_0^{t^2} (3x^2 + 1) \; dx[/math].
• Find [math]F^\prime(t)[/math] and [math]F^\prime(2)[/math] by taking the derivative of the answer to \ref{ex4.5.9a}.
• Find [math]F^\prime(t)[/math] directly using just the Fundamental Theorem and the Chain Rule.

If [math]f[/math] is continuous and [math]g[/math] is differentiable and if [math]F(t) = \int_a^{g(t)} f(x) \; dx[/math], use the Fundamental Theorem and the Chain Rule to show that [math]F^\prime (t) = f(g(t))g^\prime(t)[/math].