Evaluate
6-\sin(2)\approx 5.090702573
Share
Copied to clipboard
\int 3x-\cos(x)\mathrm{d}x
Evaluate the indefinite integral first.
\int 3x\mathrm{d}x+\int -\cos(x)\mathrm{d}x
Integrate the sum term by term.
3\int x\mathrm{d}x-\int \cos(x)\mathrm{d}x
Factor out the constant in each of the terms.
\frac{3x^{2}}{2}-\int \cos(x)\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 3 times \frac{x^{2}}{2}.
\frac{3x^{2}}{2}-\sin(x)
Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.
\frac{3}{2}\times 2^{2}-\sin(2)-\left(\frac{3}{2}\times 0^{2}-\sin(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.
6-\sin(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}