Упр.10.3 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1. $$\int \sin 5x\,dx=-\frac{\cos 5x}{5}+C.$$
2. $$\int 2\cos \frac{x}{2}\,dx=2\cdot 2\sin \frac{x}{2}+C=4\sin \frac{x}{2}+C.$$
3. $$\int \left(6x+\frac12\right)^3dx=\frac16\int \left(6x+\frac12\right)^3\,d(6x+\tfrac12)=\frac{1}{24}\left(6x+\frac12\right)^4+C.$$
4. $$\int \left(\frac{x}{7}-2\right)^4dx=7\int \left(\frac{x}{7}-2\right)^4\,d\left(\frac{x}{7}-2\right)=\frac{7}{5}\left(\frac{x}{7}-2\right)^5+C.$$
5. $$\int \frac{1}{e^{2x}}\,dx=\int e^{-2x}\,dx=-\frac12 e^{-2x}+C=-\frac{1}{2e^{2x}}+C.$$
6. $$\int 7^{3x}\,dx=\frac{1}{3\ln 7}\,7^{3x}+C.$$
7. $$\int -\frac13\sin\left(\frac{x}{3}-\frac{\pi}{4}\right)dx=\cos\left(\frac{x}{3}-\frac{\pi}{4}\right)+C.$$
8. На промежутке $$\left(-\frac{\pi}{6};\frac{\pi}{6}\right)$$
   $$\int \frac{1}{\cos^2 3x}\,dx=\int \sec^2 3x\,dx=\frac13\tan 3x+C.$$
9. На промежутке $$\left(0;\frac{\pi}{4}\right)$$
   $$\int \frac{8}{\sin^2 4x}\,dx=8\int \csc^2 4x\,dx=-2\cot 4x+C.$$
10. На промежутке $$\left(\frac12;+\infty\right)$$
    $$\int \frac{1}{\sqrt{2x-1}}\,dx=\int (2x-1)^{-\frac12}dx=\sqrt{2x-1}+C.$$
11. На промежутке $$[4;+\infty)$$
    $$\int \sqrt{x+4}\,dx=\int (x+4)^{\frac12}dx=\frac23(x+4)^{\frac32}+C.$$
12. На промежутке $$\left(-\frac23;+\infty\right)$$
    $$\int \frac{6}{3x+2}\,dx=2\ln|3x+2|+C.$$
13. На промежутке $$\left(-\infty;\frac34\right)$$
    $$\int \frac{4}{(4x-3)^2}\,dx=\int 4(4x-3)^{-2}dx=-\frac{1}{4x-3}+C.$$
14. На промежутке $$(-\infty;2]$$
    $$\int \sqrt{1-\frac{x}{2}}\,dx=-\frac43\left(1-\frac{x}{2}\right)^{\frac32}+C.$$

Ответ

1. $$F(x)=-\frac{\cos 5x}{5}+C$$
2. $$F(x)=4\sin \frac{x}{2}+C$$
3. $$F(x)=\frac{1}{24}\left(6x+\frac12\right)^4+C$$
4. $$F(x)=\frac75\left(\frac{x}{7}-2\right)^5+C$$
5. $$F(x)=-\frac{1}{2e^{2x}}+C$$
6. $$F(x)=\frac{7^{3x}}{3\ln 7}+C$$
7. $$F(x)=\cos\left(\frac{x}{3}-\frac{\pi}{4}\right)+C$$
8. $$F(x)=\frac13\tan 3x+C$$
9. $$F(x)=-2\cot 4x+C$$
10. $$F(x)=\sqrt{2x-1}+C$$
11. $$F(x)=\frac23(x+4)^{\frac32}+C$$
12. $$F(x)=2\ln|3x+2|+C$$
13. $$F(x)=-\frac{1}{4x-3}+C$$
14. $$F(x)=-\frac43\left(1-\frac{x}{2}\right)^{\frac32}+C$$