giúp mình với

Bài 1. 1) $\int(2x+1)dx = \int 2x dx + \int 1 dx = x^2 + x + C$ 2) $\int(x^2+x+2)dx = \int x^2 dx + \int x dx + \int 2 dx = \frac{x^3}{3} + \frac{x^2}{2} + 2x + C$ 3) $\int(x+1)^2dx = \int(x^2 + 2x + 1)dx = \int x^2 dx + \int 2x dx + \int 1 dx = \frac{x^3}{3} + x^2 + x + C$ 4) $\int x(x+1)^3dx = \int x(x^3 + 3x^2 + 3x + 1)dx = \int (x^4 + 3x^3 + 3x^2 + x)dx = \frac{x^5}{5} + \frac{3x^4}{4} + x^3 + \frac{x^2}{2} + C$ 5) $\int(x^2-3x+\frac{1}{x})dx = \int x^2 dx - \int 3x dx + \int \frac{1}{x} dx = \frac{x^3}{3} - \frac{3x^2}{2} + \ln|x| + C$ 6) $\int(\frac{1}{4}x^4+2x^3+\sqrt{x}+x)dx = \int \frac{1}{4}x^4 dx + \int 2x^3 dx + \int \sqrt{x} dx + \int x dx = \frac{x^5}{20} + \frac{x^4}{2} + \frac{2x^{\frac{3}{2}}}{3} + \frac{x^2}{2} + C$ 7) $\int(e^x+\sqrt[3]{x^2})dx = \int e^x dx + \int x^{\frac{2}{3}} dx = e^x + \frac{3x^{\frac{5}{3}}}{5} + C$ 8) $\int(2\sin x+\cos x)dx = \int 2\sin x dx + \int \cos x dx = -2\cos x + \sin x + C$ 9) $\int(\frac{1}{\sqrt{x}}+\frac{1}{x^2}+\frac{1}{\sqrt[3]{x}})dx = \int x^{-\frac{1}{2}} dx + \int x^{-2} dx + \int x^{-\frac{1}{3}} dx = 2x^{\frac{1}{2}} - \frac{1}{x} + \frac{3x^{\frac{2}{3}}}{2} + C$ 10) $\int(x+1)^2(x+3)dx = \int(x^2 + 2x + 1)(x + 3)dx = \int(x^3 + 5x^2 + 7x + 3)dx = \frac{x^4}{4} + \frac{5x^3}{3} + \frac{7x^2}{2} + 3x + C$ 11) $\int\frac{x^2+1}{x}dx = \int(x + \frac{1}{x})dx = \int x dx + \int \frac{1}{x} dx = \frac{x^2}{2} + \ln|x| + C$ 12) $\int\frac{x^4+2x^3+x^2+x}{x^2}dx = \int(x^2 + 2x + 1 + \frac{1}{x})dx = \int x^2 dx + \int 2x dx + \int 1 dx + \int \frac{1}{x} dx = \frac{x^3}{3} + x^2 + x + \ln|x| + C$ 13) $\int(2^x+3^x+5^x)dx = \int 2^x dx + \int 3^x dx + \int 5^x dx = \frac{2^x}{\ln 2} + \frac{3^x}{\ln 3} + \frac{5^x}{\ln 5} + C$ 14) $\int(2^{2x}.5^{3x}.7^x)dx = \int(2^x)^2.(5^x)^3.7^x dx = \int(2^x)^2.(5^x)^3.7^x dx = \int(2^x)^2.(5^x)^3.7^x dx = \int(2^x)^2.(5^x)^3.7^x dx = \frac{(2^x)^2.(5^x)^3.7^x}{\ln(2^2.5^3.7)} + C$ 15) $\int(e^x+\sin x+\frac{1}{\cos^2x})dx = \int e^x dx + \int \sin x dx + \int \sec^2 x dx = e^x - \cos x + \tan x + C$ Bài 2. 1) $\int \sin 5x dx = -\frac{1}{5}\cos 5x + C$ 2) $\int \cos 3x dx = \frac{1}{3}\sin 3x + C$ 3) $\int \sin (2x+1) dx = -\frac{1}{2}\cos (2x+1) + C$ 4) $\int \frac{1}{\cos^2(2x-2)} dx = \frac{1}{2}\tan (2x-2) + C$ 5) $\int \frac{1}{\sin^2 4x} dx = -\frac{1}{4}\cot 4x + C$ 6) $\int (2x+1)^5 dx = \frac{1}{12}(2x+1)^6 + C$ 7) $\int (x-1)^{10} dx = \frac{1}{11}(x-1)^{11} + C$ 8) $\int \sqrt{x-1} dx = \frac{2}{3}(x-1)^{\frac{3}{2}} + C$ 9) $\int \sqrt{3x+2} dx = \frac{2}{9}(3x+2)^{\frac{3}{2}} + C$ 10) $\int \sqrt{(x-5)^4} dx = \int |x-5|^2 dx = \int (x-5)^2 dx = \frac{1}{3}(x-5)^3 + C$ 11) $\int \frac{1}{\sqrt{7x-5}} dx = \frac{2}{7}\sqrt{7x-5} + C$ 12) $\int \frac{1}{\sqrt{(\frac{1}{2}x+2)^3}} dx = \int (\frac{1}{2}x+2)^{-\frac{3}{2}} dx = -4(\frac{1}{2}x+2)^{-\frac{1}{2}} + C$ 13) $\int e^{2x+4} dx = \frac{1}{2}e^{2x+4} + C$ 14) $\int (\frac{1}{2x+1} + \frac{2}{3x-2}) dx = \frac{1}{2}\ln|2x+1| + \frac{2}{3}\ln|3x-2| + C$ 15) $\int (e^{3x+1} + \sqrt[3]{(2x+1)^2} + \frac{1}{2x+1}) dx = \frac{1}{3}e^{3x+1} + \frac{3}{8}(2x+1)^{\frac{5}{3}} + \frac{1}{2}\ln|2x+1| + C$ Đáp số: 1) $-\frac{1}{5}\cos 5x + C$ 2) $\frac{1}{3}\sin 3x + C$ 3) $-\frac{1}{2}\cos (2x+1) + C$ 4) $\frac{1}{2}\tan (2x-2) + C$ 5) $-\frac{1}{4}\cot 4x + C$ 6) $\frac{1}{12}(2x+1)^6 + C$ 7) $\frac{1}{11}(x-1)^{11} + C$ 8) $\frac{2}{3}(x-1)^{\frac{3}{2}} + C$ 9) $\frac{2}{9}(3x+2)^{\frac{3}{2}} + C$ 10) $\frac{1}{3}(x-5)^3 + C$ 11) $\frac{2}{7}\sqrt{7x-5} + C$ 12) $-4(\frac{1}{2}x+2)^{-\frac{1}{2}} + C$ 13) $\frac{1}{2}e^{2x+4} + C$ 14) $\frac{1}{2}\ln|2x+1| + \frac{2}{3}\ln|3x-2| + C$ 15) $\frac{1}{3}e^{3x+1} + \frac{3}{8}(2x+1)^{\frac{5}{3}} + \frac{1}{2}\ln|2x+1| + C$ Bài 3. 1) $\int \frac{x}{x^2 + 1} dx$ Đặt $u = x^2 + 1$, suy ra $du = 2x dx$. Do đó: \[ \int \frac{x}{x^2 + 1} dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln |u| + C = \frac{1}{2} \ln |x^2 + 1| + C \] 2) $\int \frac{x^2}{x^3 + 10} dx$ Đặt $u = x^3 + 10$, suy ra $du = 3x^2 dx$. Do đó: \[ \int \frac{x^2}{x^3 + 10} dx = \frac{1}{3} \int \frac{du}{u} = \frac{1}{3} \ln |u| + C = \frac{1}{3} \ln |x^3 + 10| + C \] 3) $\int (1 + x^4)^3 x^3 dx$ Đặt $u = 1 + x^4$, suy ra $du = 4x^3 dx$. Do đó: \[ \int (1 + x^4)^3 x^3 dx = \frac{1}{4} \int u^3 du = \frac{1}{4} \cdot \frac{u^4}{4} + C = \frac{1}{16} (1 + x^4)^4 + C \] 4) $\int \frac{x}{\sqrt{x^2 + 4}} dx$ Đặt $u = x^2 + 4$, suy ra $du = 2x dx$. Do đó: \[ \int \frac{x}{\sqrt{x^2 + 4}} dx = \frac{1}{2} \int \frac{du}{\sqrt{u}} = \frac{1}{2} \cdot 2 \sqrt{u} + C = \sqrt{x^2 + 4} + C \] 5) $\int e^{1 + \sin x} \cos x dx$ Đặt $u = 1 + \sin x$, suy ra $du = \cos x dx$. Do đó: \[ \int e^{1 + \sin x} \cos x dx = \int e^u du = e^u + C = e^{1 + \sin x} + C \] 6) $\int e^{x^2 + 1} x dx$ Đặt $u = x^2 + 1$, suy ra $du = 2x dx$. Do đó: \[ \int e^{x^2 + 1} x dx = \frac{1}{2} \int e^u du = \frac{1}{2} e^u + C = \frac{1}{2} e^{x^2 + 1} + C \] 7) $\int \frac{e^{\frac{1}{x}}}{x^2} dx$ Đặt $u = \frac{1}{x}$, suy ra $du = -\frac{1}{x^2} dx$. Do đó: \[ \int \frac{e^{\frac{1}{x}}}{x^2} dx = -\int e^u du = -e^u + C = -e^{\frac{1}{x}} + C \] 8) $\int \frac{1}{x \ln^4 x} dx$ Đặt $u = \ln x$, suy ra $du = \frac{1}{x} dx$. Do đó: \[ \int \frac{1}{x \ln^4 x} dx = \int \frac{du}{u^4} = \int u^{-4} du = \frac{u^{-3}}{-3} + C = -\frac{1}{3 \ln^3 x} + C \] 9) $\int \frac{(5 + \ln^2 x)^4 \ln x}{x} dx$ Đặt $u = 5 + \ln^2 x$, suy ra $du = 2 \ln x \cdot \frac{1}{x} dx$. Do đó: \[ \int \frac{(5 + \ln^2 x)^4 \ln x}{x} dx = \frac{1}{2} \int u^4 du = \frac{1}{2} \cdot \frac{u^5}{5} + C = \frac{1}{10} (5 + \ln^2 x)^5 + C \]