Solve ∫ x(2tx^2)^2 wrt x

$$\int x ( 2 t x ^ { 2 } ) ^ { 2 } d x$$

Evaluate

$$\frac{2t^{2}x^{6}}{3}+С$$

Solution Steps

Expand \left(2tx^{2}\right)^{2}.

$$\int x\times 2^{2}t^{2}\left(x^{2}\right)^{2}\mathrm{d}x$$

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

$$\int x\times 2^{2}t^{2}x^{4}\mathrm{d}x$$

Calculate 2 to the power of 2 and get 4.

$$\int x\times 4t^{2}x^{4}\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\int x^{5}\times 4t^{2}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$4t^{2}\int x^{5}\mathrm{d}x$$

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}.

$$4t^{2}\times \frac{x^{6}}{6}$$

Simplify.

$$\frac{2t^{2}x^{6}}{3}$$

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.

$$\frac{2t^{2}x^{6}}{3}+С$$

Differentiate w.r.t. x

$$4t^{2}x^{5}$$