Evaluate
-\frac{27}{2}=-13.5
Share
Copied to clipboard
\int t^{2}-t-6\mathrm{d}t
Evaluate the indefinite integral first.
\int t^{2}\mathrm{d}t+\int -t\mathrm{d}t+\int -6\mathrm{d}t
Integrate the sum term by term.
\int t^{2}\mathrm{d}t-\int t\mathrm{d}t+\int -6\mathrm{d}t
Factor out the constant in each of the terms.
\frac{t^{3}}{3}-\int t\mathrm{d}t+\int -6\mathrm{d}t
Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.
\frac{t^{3}}{3}-\frac{t^{2}}{2}+\int -6\mathrm{d}t
Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply -1 times \frac{t^{2}}{2}.
\frac{t^{3}}{3}-\frac{t^{2}}{2}-6t
Find the integral of -6 using the table of common integrals rule \int a\mathrm{d}t=at.
\frac{3^{3}}{3}-\frac{3^{2}}{2}-6\times 3-\left(\frac{0^{3}}{3}-\frac{0^{2}}{2}-6\times 0\right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
-\frac{27}{2}
Simplify.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}