Exercise 10.1

Evaluate the following integrals.

(a) $\int 4x^8 dx$

$$ = 4\int x^8 dx = 4\frac{x^{8+1}}{8+1} + C = \frac{4}{9}x^9 + C $$

(b) $\int \frac{3}{2}x^{\frac{3}{2}}\sqrt[3]{x} dx$

$$ = \frac{3}{2}\int x^{\frac{7}{3}} dx = \frac{3}{2}\frac{x^{\frac{7}{3}+1}}{\frac{7}{3}+1} + C = \frac{9}{20}x^{\frac{10}{3}} + C $$

(c) $\int (5^x + 2) dx$

$$ = \int 5^x dx + \int 2 dx = \frac{5^x}{\ln 5} + 2x + C $$

(d) $\int \sin^2 x dx$

$$ = \int\frac{1-\cos 2x}{2} dx = \frac{1}{2}\int(1-\cos 2x) dx = \frac{1}{2}(x - \frac{\sin 2x}{2}) + C $$

(e) $\int \frac{x+3}{\sqrt{x}} dx$

$$ = \int (x^{\frac{1}{2}} + 3x^{-\frac{1}{2}}) dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + 3\frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = \frac{2}{3}x^{\frac{3}{2}} + 6x^{\frac{1}{2}} + C $$

(f) $\int (\frac{1}{x} + 5) dx$

$$ = \int \frac{1}{x} dx + 5\int dx = \ln|x| + 5x + C $$

(g) $\int (e^x + \frac{2}{x}) dx$

$$ = \int e^x dx + 2\int\frac{1}{x} dx = e^x + 2\ln|x| + C $$

(h) $\int (\frac{1}{x^5} + 4e^x) dx$

$$ = \int x^{-5} dx + 4\int e^x dx = \frac{x^{-5+1}}{-5+1} + 4e^x + C = -\frac{1}{4x^4} + 4e^x + C $$

(i) $\int (\frac{3}{x} + e^x + 10) dx$

$$ = 3\int\frac{1}{x} dx + \int e^x dx + 10\int dx = 3\ln|x| + e^x + 10x + C $$

(j) $\int \sin^2 3x dx$

$$ = \int\frac{1-\cos(6x)}{2}dx = \frac{1}{2}\int(1-\cos 6x)dx = \frac{1}{2}(x - \frac{1}{6}\sin 6x) + C = \frac{1}{2}x - \frac{1}{12}\sin 6x + C $$

(k) $\int \sin 5x \sin 2x dx$

$$ = -\frac{1}{2}\int(\cos 7x - \cos 3x)dx = -\frac{1}{2}(\frac{\sin 7x}{7} - \frac{\sin 3x}{3}) + C = \frac{\sin 3x}{6} - \frac{\sin 7x}{14} + C $$

(l) $\int \cos 7x \cos 4x dx$

$$ = \frac{1}{2}\int(\cos 11x + \cos 3x)dx = \frac{1}{2}(\frac{\sin 11x}{11} + \frac{\sin 3x}{3}) + C = \frac{\sin 11x}{22} + \frac{\sin 3x}{6} + C $$

Problem 2

Evaluate the following integrals.

(a) $\int (1-2x)^3 dx$

$$ = -\frac{1}{2}\frac{(1-2x)^{3+1}}{3+1} + C = -\frac{1}{8}(1-2x)^4 + C $$

(b) $\int \sin(2\pi x + 7) dx$

$$ = \frac{1}{2\pi}(-\cos(2\pi x + 7)) + C = -\frac{1}{2\pi}\cos(2\pi x + 7) + C $$

(c) $\int \cos(3x-7) dx$

$$ = \frac{1}{3}\sin(3x-7) + C $$

(d) $\int 3^{5x-2} dx$

$$ = \frac{3^{5x-2}}{5\ln 3} + C $$

(e) $\int \frac{1}{7x-6} dx$

$$ = \frac{1}{7}\ln|7x-6| + C $$

(f) $\int \frac{\sin 2x}{\sin x} dx$

$$ = \int \frac{2\sin x \cos x}{\sin x} dx = 2\int\cos x dx = 2\sin x + C $$

(g) $\int \sec^2(2x+3) dx$

$$ = \frac{1}{2}\tan(2x+3) + C $$

(h) $\int e^{7x-3} dx$

$$ = \frac{1}{7}e^{7x-3} + C $$

(i) $\int (1 + \tan^2 2\pi x) dx$

$$ = \int\sec^2 2\pi x dx = \frac{1}{2\pi}\tan 2\pi x + C $$