Integrals
Integration by parts
Reverse the product rule: ∫u dv=uv−∫v du. Choose u so that differentiating it simplifies the remaining integral.
1. A polynomial times an exponential
1 / 3f=xe^x,\quad F=(x-1)e^x For ∫xeˣ dx, choose u=x and dv=eˣdx. Then du=dx and v=eˣ. The result is xeˣ−eˣ+C=(x−1)eˣ+C.
- f(a)
- 2.718282
- F(a)
- 0
The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.
2. Integrate the logarithm
2 / 3f=\ln x,\quad F=x\ln x-x View ln(x) as ln(x)×1. Choose u=ln(x), dv=dx, so du=dx/x and v=x. The primitive is xln(x)−x+C on x>0.
- f(a)
- 0
- F(a)
- -1
The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.
3. Keep the boundary term
3 / 3\int_0^b x\cos x\,dx=b\sin b+\cos b-1 For ∫₀ᵇ x cos(x) dx, take u=x and dv=cos(x)dx. The result is b sin(b)+cos(b)−1. The lower endpoint contributes the −1.
- b sin(b)
- 0.841471
- ∫
- 0.381773
Shading shows the region between the bounds. With bounds in increasing order, regions below the axis contribute negatively; reversing the bounds reverses the integral’s sign. Geometric area stays nonnegative.