7.2, #12. Set w = −x^{2}, dw = −2x dx, then

∫ xe^{−x^{2}} dx = −\frac{1}{2} ∫ e^{w} dw = −\frac{1}{2}e^{w} + C = −\frac{1}{2}e^{−x^{2}} + C.

7.2, #16. Set w = y + 5, dw = dy, then

∫ \frac{1}{y + 5} dx = ∫ \frac{1}{w} dw = ln w = ln(y + 5).

7.2, #20. Set w = 4 − x, dw = −dx, then

∫ \frac{dx}{\sqrt{4 − x}} = − ∫ w^{−1/2} dw = −2w^{\frac{1}{2}} + C = −2\sqrt{4 − x} + C.

7.2, #24. Set w = cos 3t, dw = −3 sin 3t dt, then

∫ \sqrt{cos 3t} sin 3t dt = −\frac{1}{3} ∫ w^{1/2} dw = −\frac{2}{9}w^{3/2} + C = −\frac{2}{9}(cos^{3/2} 3t) + C.

7.2, #28. Set w = \sqrt{x}, dw = 1/(2\sqrt{x}) dx, then

∫ \begin{matrix}cos & \sqrt{x} \\ √ & \overline{x} \end{matrix} dx = 2 ∫ cos w dw = 2 sin w + C = 2 sin \sqrt{x} + C.

7.3, #1(a). Set w = 1 + x^{2}, dw = 2x dx. Then w = 1 if x = 0, w = 2 if x = 1, and

∫_{0}^{1} \frac{x}{1 + x^{2}} dx = \frac{1}{2} ∫_{1}^{2} \frac{dw}{w} = \frac{1}{2} ln |w||_{1}^{2} = \frac{1}{2} ln 2 = 0.347

#1(b). Set w = cos x, dw = − sin x dx. Then w = 1 if x = 0, w = 1/\sqrt{2} if x = π/4, and

∫_{0}^{π/4} \frac{sin x}{cos x} dx = − ∫_{1}^{1/\sqrt{2}} \frac{dw}{w} = − ln |w||_{1}^{1/\sqrt{2}} = − ln(1/\sqrt{2}) = 0.347.

The answers are the same since − ln(1/\sqrt{2}) = ln \sqrt{2} = (1/2) ln 2. 7.3, #2. Set w = πx, dw = πdx. Then w = 0 if x = 0, w = π/2 if x = 1/2, and

∫_{0}^{1/2} cos πx dx = \frac{1}{π} ∫_{0}^{π/2} cos w dw = \frac{1}{π} sin w|_{0}^{π/2} = \frac{1}{π} = 0.318.

7.3, #3. Set w = x^{1/3}, dw = (1/3)x^{−2/3} dx. Then w = 1 if x = 1, w = 2 if x = 8, and

∫_{1}^{8} \frac{e^{\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} dx = 3 ∫_{1}^{2} e^{w} dw = 3e^{w}|_{1}^{2} = 3(e^{2} − e) = 14.012.