Evaluate the following indefinite integrals.

• [math]\int (x^2 + x + 1) \; dx[/math]
• [math]\int \left(3x^2 - \frac1{3x^3}\right) \; dx[/math]
• [math]\int (6t^2 - 2t + 5) \; dt[/math]
• [math]\int (2y + 1)(y - 3) \; dy[/math]
• [math]\int (2x-1)^\frac32 \; dx[/math]
• [math]\int (3x^3 + 2)^5x^2 \; dx[/math]
• [math]\int x\sqrt{a^2 - x^2} \; dx[/math]
• [math]\int \frac{t + 2}{\sqrt{t^2 + 4t + 5}} \; dt[/math]
• [math]\int s(s^3 + 3s^2 + 5)(s + 2) \; ds[/math]
• [math]\int |x| \; dx[/math].

Among the following integrals identify those that can be evaluated using the techniques in this section. Evaluate them.

• [math]\int \left(x + \frac1x\right) \; dx[/math]
• [math]\int \left(\sqrt{x} + \frac1{\sqrt{x}}\right) \; dx[/math]
• [math]\int y^2(y^3 + 7)^4 \; dy[/math]
• [math]\int y(y^3 + 7)^4 \; dy[/math]
• [math]\int t\sqrt{t^3 - 1} \; dt[/math]
• [math]\int \frac{x+1}{x-1} \; dx[/math]
• [math]\int (3x^2 - 1)(x + 2) \; dx[/math]
• [math]\int (s+1)(s^2 + 2s - 3)^4 \; ds[/math]
• [math]\int \frac{x^2 - 1}{x + 1} \; dx[/math]
• [math]\int \frac{y + 2}{y^2 + 1} \; dy[/math]
• [math]\int \frac{x - 1}{(x + 1)^3} \; dx[/math].

The curve defined by [math]y = f(x)[/math] passes through the point [math](1,4)[/math]. In addition, at each point [math](x,f(x))[/math], the slope of the curve is [math]8x^3 + 2x[/math]. Find [math]f(x)[/math].

The line tangent to the graph of the differentiable function [math]f[/math] at each point [math](x,f(x))[/math] has slope [math]3x^2 + 1[/math], and the graph passes through the point [math](2,9)[/math]. Find [math]f(x)[/math].

If [math]f^{\prime\prime}(x) = 12x^2 + 2[/math] and the graph of [math]y = f(x)[/math] passes through [math](0,-2)[/math] with a slope of [math]5[/math], find [math]f(x)[/math].

Evaluate the following definite integrals.

• [math]\int_0^1 (3x^2 + 4x + 1) \; dx[/math]
• [math]\int_{-1}^1 (2t^3 + t) \; dt[/math]
• [math]\int_{-1}^1 (x^3 + 1)^{17}x^2 \; dx[/math]
• [math]\int_{-1}^2 \frac{s+1}{\sqrt{s^2 + 2s + 3}} \; ds[/math]
• [math]\int_1^3 \left(x^2 + \frac1x\right)^3 \left(2x - \frac1{x^2} \right) \; dx[/math]
• [math]\int_0^2 \frac1{(x+1)^2} \; dx[/math]
• [math]\int_{-2}^2 \sqrt{4-x^2} \, x \; dx[/math]
• [math]\int_{-2}^2 (2|x| + 1) \; dx[/math]
• [math]\int_0^1 t(t^3 + 3t^2 - 1)^3 (t + 2) \; dt[/math]
• [math]\int_1^2 \frac{x^4 + 2x^3 - 2}{x^2} \; dx[/math].

If [math]f^{\prime\prime}(x) = 18x + 10[/math] and [math]f^\prime(0) = 2[/math], find [math]f^\prime(x)[/math]. If, in addition, [math]f(0) = 1[/math], find [math]f(x)[/math].

• If [math]g^{\prime\prime}(x) = \sqrt x[/math] and [math]g^\prime (1) = 0[/math] and [math]g(0) = \sqrt2[/math], find [math]g(x)[/math].
• If [math]f^{\prime\prime\prime}(t) = 6[/math] and [math]f^{\prime\prime}(1) = 8[/math] and [math]f^\prime(0) = 1[/math] and [math]f(1) = 4[/math], find [math]f(t)[/math].

If the slope of the curve [math]y = f(x)[/math] is equal to [math]6[/math] at the point [math](1,4)[/math] and, more generally, equals [math]6x[/math] at [math](x,f(x))[/math], what is the area bounded by the curve [math]y = f(x)[/math], the [math]x[/math]-axis, and the lines [math]x = 1[/math] and [math]x = 3[/math]?

Sketch the region bounded by the curves [math]y = \frac1{\sqrt{x + 1}}[/math], [math]x = 0[/math], and [math]y = \frac12[/math]. Find its area.