Integration by Substitution
Use u=x^2+c on x in [0,b] to evaluate integral_0^b 2x(x^2+c)^p dx. Show original and transformed integrands, linked bounds, and equal integral readouts.
Free theory articleUse u=x^2+c on x in [0,b] to evaluate integral_0^b 2x(x^2+c)^p dx. Show original and transformed integrands, linked bounds, and equal integral readouts.
Concept & interpretation
Integration by substitution, also known as u-substitution, is a method for evaluating integrals that reverses the chain rule. By choosing an appropriate substitution u = g(x), we can transform a complex integral into a simpler one, often mapping the integration bounds concurrently.
Experiment
Explore the transformation in the Integration by Substitution visualization:
- Set the Upper bound b to
1, Shift c to1, and Power p to1. - Observe that the transformed u bounds shift to [1, 2] and the integral evaluates to 1.5.
- Set the Upper bound b to
2, Shift c to0, and Power p to2. - The u bounds become [0, 4] and the integral evaluates exactly to 64/3, or approximately 21.33333.
- Set Upper bound b = 1, Shift c = 0 and Power p = 0. The original integrand is 2x, the transformed integrand is 1 and both integral readouts are 1.
- Set Upper bound b = 0, Shift c = 1 and Power p = 2. Both intervals collapse and both readouts become 0.
- Set Upper bound b = 3, Shift c = 2 and Power p = 3. Both readouts are 3656.25, from (114 − 24)/4.
- Clear Upper bound b. The visible error replaces all dependent calculations. Restore 1, then pan and choose Reset view; parameter values stay unchanged.