Solve ∫ (3sint+1)dt

$$\int ( 3 \sin t + 1 ) d t$$

Evaluate

$$-3\cos(t)+t+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3\sin(t)\mathrm{d}t+\int 1\mathrm{d}t$$

Factor out the constant in each of the terms.

$$3\int \sin(t)\mathrm{d}t+\int 1\mathrm{d}t$$

Use \int \sin(t)\mathrm{d}t=-\cos(t) from the table of common integrals to obtain the result. Multiply 3 times -\cos(t).

$$-3\cos(t)+\int 1\mathrm{d}t$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}t=at.

$$-3\cos(t)+t$$

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.

$$-3\cos(t)+t+С$$

Differentiate w.r.t. t

$$3\sin(t)+1$$