Integral of (x+3)^20 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = x + 3$.

      Then let $du = dx$ and substitute $du$:

      $$\int u^{20}\, du$$

      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int u^{20}\, du = \frac{u^{21}}{21}$$

      Now substitute $u$ back in:

      $$\frac{\left(x + 3\right)^{21}}{21}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(x + 3\right)^{20} = x^{20} + 60 x^{19} + 1710 x^{18} + 30780 x^{17} + 392445 x^{16} + 3767472 x^{15} + 28256040 x^{14} + 169536240 x^{13} + 826489170 x^{12} + 3305956680 x^{11} + 10909657044 x^{10} + 29753610120 x^{9} + 66945622770 x^{8} + 123591918960 x^{7} + 185387878440 x^{6} + 222465454128 x^{5} + 208561363245 x^{4} + 147219785820 x^{3} + 73609892910 x^{2} + 23245229340 x + 3486784401$$

   2. Integrate term-by-term:

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int x^{20}\, dx = \frac{x^{21}}{21}$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 60 x^{19}\, dx = 60 \int x^{19}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{19}\, dx = \frac{x^{20}}{20}$$

         So, the result is: $3 x^{20}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 1710 x^{18}\, dx = 1710 \int x^{18}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{18}\, dx = \frac{x^{19}}{19}$$

         So, the result is: $90 x^{19}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 30780 x^{17}\, dx = 30780 \int x^{17}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{17}\, dx = \frac{x^{18}}{18}$$

         So, the result is: $1710 x^{18}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 392445 x^{16}\, dx = 392445 \int x^{16}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{16}\, dx = \frac{x^{17}}{17}$$

         So, the result is: $23085 x^{17}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 3767472 x^{15}\, dx = 3767472 \int x^{15}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{15}\, dx = \frac{x^{16}}{16}$$

         So, the result is: $235467 x^{16}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 28256040 x^{14}\, dx = 28256040 \int x^{14}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{14}\, dx = \frac{x^{15}}{15}$$

         So, the result is: $1883736 x^{15}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 169536240 x^{13}\, dx = 169536240 \int x^{13}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{13}\, dx = \frac{x^{14}}{14}$$

         So, the result is: $\frac{84768120 x^{14}}{7}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 826489170 x^{12}\, dx = 826489170 \int x^{12}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{12}\, dx = \frac{x^{13}}{13}$$

         So, the result is: $63576090 x^{13}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 3305956680 x^{11}\, dx = 3305956680 \int x^{11}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{11}\, dx = \frac{x^{12}}{12}$$

         So, the result is: $275496390 x^{12}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 10909657044 x^{10}\, dx = 10909657044 \int x^{10}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{10}\, dx = \frac{x^{11}}{11}$$

         So, the result is: $991787004 x^{11}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 29753610120 x^{9}\, dx = 29753610120 \int x^{9}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{9}\, dx = \frac{x^{10}}{10}$$

         So, the result is: $2975361012 x^{10}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 66945622770 x^{8}\, dx = 66945622770 \int x^{8}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

         So, the result is: $7438402530 x^{9}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 123591918960 x^{7}\, dx = 123591918960 \int x^{7}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

         So, the result is: $15448989870 x^{8}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 185387878440 x^{6}\, dx = 185387878440 \int x^{6}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{6}\, dx = \frac{x^{7}}{7}$$

         So, the result is: $\frac{185387878440 x^{7}}{7}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 222465454128 x^{5}\, dx = 222465454128 \int x^{5}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

         So, the result is: $37077575688 x^{6}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 208561363245 x^{4}\, dx = 208561363245 \int x^{4}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

         So, the result is: $41712272649 x^{5}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 147219785820 x^{3}\, dx = 147219785820 \int x^{3}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

         So, the result is: $36804946455 x^{4}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 73609892910 x^{2}\, dx = 73609892910 \int x^{2}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

         So, the result is: $24536630970 x^{3}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 23245229340 x\, dx = 23245229340 \int x\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int x\, dx = \frac{x^{2}}{2}$$

         So, the result is: $11622614670 x^{2}$

      1. The integral of a constant is the constant times the variable of integration:

         $$\int 3486784401\, dx = 3486784401 x$$

      The result is: $\frac{x^{21}}{21} + 3 x^{20} + 90 x^{19} + 1710 x^{18} + 23085 x^{17} + 235467 x^{16} + 1883736 x^{15} + \frac{84768120 x^{14}}{7} + 63576090 x^{13} + 275496390 x^{12} + 991787004 x^{11} + 2975361012 x^{10} + 7438402530 x^{9} + 15448989870 x^{8} + \frac{185387878440 x^{7}}{7} + 37077575688 x^{6} + 41712272649 x^{5} + 36804946455 x^{4} + 24536630970 x^{3} + 11622614670 x^{2} + 3486784401 x$

2. Now simplify:

   $$\frac{\left(x + 3\right)^{21}}{21}$$

3. Add the constant of integration:

   $$\frac{\left(x + 3\right)^{21}}{21}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(x + 3\right)^{21}}{21}+ \mathrm{constant}$$