Solve ∫ (2x^2+3)^3 wrt x

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

Evaluate

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+27x+С$$

Solution Steps

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

$$\int 8\left(x^{2}\right)^{3}+36\left(x^{2}\right)^{2}+54x^{2}+27\mathrm{d}x$$

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

$$\int 8x^{6}+36\left(x^{2}\right)^{2}+54x^{2}+27\mathrm{d}x$$

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

$$\int 8x^{6}+36x^{4}+54x^{2}+27\mathrm{d}x$$

Integrate the sum term by term.

$$\int 8x^{6}\mathrm{d}x+\int 36x^{4}\mathrm{d}x+\int 54x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

Factor out the constant in each of the terms.

$$8\int x^{6}\mathrm{d}x+36\int x^{4}\mathrm{d}x+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply 8 times \frac{x^{7}}{7}.

$$\frac{8x^{7}}{7}+36\int x^{4}\mathrm{d}x+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply 36 times \frac{x^{5}}{5}.

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 54 times \frac{x^{3}}{3}.

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+\int 27\mathrm{d}x$$

Find the integral of 27 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+27x$$

Simplify.

$$27x+18x^{3}+\frac{36x^{5}}{5}+\frac{8x^{7}}{7}$$

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.

$$27x+18x^{3}+\frac{36x^{5}}{5}+\frac{8x^{7}}{7}+С$$

Differentiate w.r.t. x

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