Solve ∫ (4x)(x^2+5)^3 wrt x

$$\int ( 4 x ) ( x ^ { 2 } + 5 ) ^ { 3 } d x$$

Evaluate

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+250x^{2}+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x^{2}+5\right)^{3}.

$$\int 4x\left(\left(x^{2}\right)^{3}+15\left(x^{2}\right)^{2}+75x^{2}+125\right)\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\int 4x\left(x^{6}+15\left(x^{2}\right)^{2}+75x^{2}+125\right)\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int 4x\left(x^{6}+15x^{4}+75x^{2}+125\right)\mathrm{d}x$$

Use the distributive property to multiply 4x by x^{6}+15x^{4}+75x^{2}+125.

$$\int 4x^{7}+60x^{5}+300x^{3}+500x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{7}\mathrm{d}x+\int 60x^{5}\mathrm{d}x+\int 300x^{3}\mathrm{d}x+\int 500x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{7}\mathrm{d}x+60\int x^{5}\mathrm{d}x+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}. Multiply 4 times \frac{x^{8}}{8}.

$$\frac{x^{8}}{2}+60\int x^{5}\mathrm{d}x+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 60 times \frac{x^{6}}{6}.

$$\frac{x^{8}}{2}+10x^{6}+300\int x^{3}\mathrm{d}x+500\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 300 times \frac{x^{4}}{4}.

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+500\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 500 times \frac{x^{2}}{2}.

$$\frac{x^{8}}{2}+10x^{6}+75x^{4}+250x^{2}$$

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.

$$250x^{2}+75x^{4}+10x^{6}+\frac{x^{8}}{2}+С$$

Differentiate w.r.t. x

$$4x\left(x^{2}+5\right)^{3}$$