Solve ∫ (t^4+x^3+2)dt

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

Evaluate

$$tx^{3}+\frac{t^{5}}{5}+2t+С$$

Solution Steps

Integrate the sum term by term.

$$\int t^{4}\mathrm{d}t+\int x^{3}\mathrm{d}t+\int 2\mathrm{d}t$$

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

$$\frac{t^{5}}{5}+\int x^{3}\mathrm{d}t+\int 2\mathrm{d}t$$

Find the integral of x^{3} using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{t^{5}}{5}+x^{3}t+\int 2\mathrm{d}t$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{t^{5}}{5}+x^{3}t+2t$$

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

$$\frac{t^{5}}{5}+x^{3}t+2t+С$$

Differentiate w.r.t. x

$$3tx^{2}$$