Solve ∫ (x^2+x^2)^7(3x^2+2x) wrt x

$$\int ( x ^ { 2 } + x ^ { 2 } ) ^ { 7 } ( 3 x ^ { 2 } + 2 x ) d x$$

Evaluate

$$\frac{384x^{17}}{17}+16x^{16}+С$$

Solution Steps

Combine x^{2} and x^{2} to get 2x^{2}.

$$\int \left(2x^{2}\right)^{7}\left(3x^{2}+2x\right)\mathrm{d}x$$

Expand \left(2x^{2}\right)^{7}.

$$\int 2^{7}\left(x^{2}\right)^{7}\left(3x^{2}+2x\right)\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 7 to get 14.

$$\int 2^{7}x^{14}\left(3x^{2}+2x\right)\mathrm{d}x$$

Calculate 2 to the power of 7 and get 128.

$$\int 128x^{14}\left(3x^{2}+2x\right)\mathrm{d}x$$

Use the distributive property to multiply 128x^{14} by 3x^{2}+2x.

$$\int 384x^{16}+256x^{15}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 384x^{16}\mathrm{d}x+\int 256x^{15}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$384\int x^{16}\mathrm{d}x+256\int x^{15}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{16}\mathrm{d}x with \frac{x^{17}}{17}. Multiply 384 times \frac{x^{17}}{17}.

$$\frac{384x^{17}}{17}+256\int x^{15}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{15}\mathrm{d}x with \frac{x^{16}}{16}. Multiply 256 times \frac{x^{16}}{16}.

$$\frac{384x^{17}}{17}+16x^{16}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{384x^{17}}{17}+16x^{16}+С$$

Differentiate w.r.t. x

$$128\left(3x+2\right)x^{15}$$