Integral of (3x+4)^10 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 3 x + 4$.

      Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

      $$\int \frac{u^{10}}{3}\, du$$

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

         $$\int u^{10}\, du = \frac{\int u^{10}\, du}{3}$$

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

            $$\int u^{10}\, du = \frac{u^{11}}{11}$$

         So, the result is: $\frac{u^{11}}{33}$

      Now substitute $u$ back in:

      $$\frac{\left(3 x + 4\right)^{11}}{33}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(3 x + 4\right)^{10} = 59049 x^{10} + 787320 x^{9} + 4723920 x^{8} + 16796160 x^{7} + 39191040 x^{6} + 62705664 x^{5} + 69672960 x^{4} + 53084160 x^{3} + 26542080 x^{2} + 7864320 x + 1048576$$

   2. Integrate term-by-term:

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

         $$\int 59049 x^{10}\, dx = 59049 \int x^{10}\, dx$$

         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: $\frac{59049 x^{11}}{11}$

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

         $$\int 787320 x^{9}\, dx = 787320 \int x^{9}\, dx$$

         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: $78732 x^{10}$

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

         $$\int 4723920 x^{8}\, dx = 4723920 \int x^{8}\, dx$$

         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: $524880 x^{9}$

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

         $$\int 16796160 x^{7}\, dx = 16796160 \int x^{7}\, dx$$

         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: $2099520 x^{8}$

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

         $$\int 39191040 x^{6}\, dx = 39191040 \int x^{6}\, dx$$

         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: $5598720 x^{7}$

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

         $$\int 62705664 x^{5}\, dx = 62705664 \int x^{5}\, dx$$

         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: $10450944 x^{6}$

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

         $$\int 69672960 x^{4}\, dx = 69672960 \int x^{4}\, dx$$

         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: $13934592 x^{5}$

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

         $$\int 53084160 x^{3}\, dx = 53084160 \int x^{3}\, dx$$

         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: $13271040 x^{4}$

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

         $$\int 26542080 x^{2}\, dx = 26542080 \int x^{2}\, dx$$

         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: $8847360 x^{3}$

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

         $$\int 7864320 x\, dx = 7864320 \int x\, dx$$

         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: $3932160 x^{2}$

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

         $$\int 1048576\, dx = 1048576 x$$

      The result is: $\frac{59049 x^{11}}{11} + 78732 x^{10} + 524880 x^{9} + 2099520 x^{8} + 5598720 x^{7} + 10450944 x^{6} + 13934592 x^{5} + 13271040 x^{4} + 8847360 x^{3} + 3932160 x^{2} + 1048576 x$

2. Now simplify:

   $$\frac{\left(3 x + 4\right)^{11}}{33}$$

3. Add the constant of integration:

   $$\frac{\left(3 x + 4\right)^{11}}{33}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(3 x + 4\right)^{11}}{33}+ \mathrm{constant}$$