Solve ∫ x(4x^2+3)^3 wrt x

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

Evaluate

$$8x^{8}+24x^{6}+27x^{4}+\frac{27x^{2}}{2}+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(4x^{2}+3\right)^{3}.

$$\int x\left(64\left(x^{2}\right)^{3}+144\left(x^{2}\right)^{2}+108x^{2}+27\right)\mathrm{d}x$$

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

$$\int x\left(64x^{6}+144\left(x^{2}\right)^{2}+108x^{2}+27\right)\mathrm{d}x$$

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

$$\int x\left(64x^{6}+144x^{4}+108x^{2}+27\right)\mathrm{d}x$$

Use the distributive property to multiply x by 64x^{6}+144x^{4}+108x^{2}+27.

$$\int 64x^{7}+144x^{5}+108x^{3}+27x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 64x^{7}\mathrm{d}x+\int 144x^{5}\mathrm{d}x+\int 108x^{3}\mathrm{d}x+\int 27x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$64\int x^{7}\mathrm{d}x+144\int x^{5}\mathrm{d}x+108\int x^{3}\mathrm{d}x+27\int x\mathrm{d}x$$

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 64 times \frac{x^{8}}{8}.

$$8x^{8}+144\int x^{5}\mathrm{d}x+108\int x^{3}\mathrm{d}x+27\int x\mathrm{d}x$$

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 144 times \frac{x^{6}}{6}.

$$8x^{8}+24x^{6}+108\int x^{3}\mathrm{d}x+27\int x\mathrm{d}x$$

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 108 times \frac{x^{4}}{4}.

$$8x^{8}+24x^{6}+27x^{4}+27\int x\mathrm{d}x$$

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 27 times \frac{x^{2}}{2}.

$$8x^{8}+24x^{6}+27x^{4}+\frac{27x^{2}}{2}$$

Simplify.

$$\frac{27x^{2}}{2}+27x^{4}+24x^{6}+8x^{8}$$

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.

$$\frac{27x^{2}}{2}+27x^{4}+24x^{6}+8x^{8}+С$$

Differentiate w.r.t. x

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