Calculus: Integrals Practice (Antiderivatives, u-Substitution and Area)

Reverse power rule: . So , plus C. is the derivative, and leaving off + C loses the family of antiderivatives.

The power rule fails for (it would divide by zero). The antiderivative of is ; the absolute value covers negative x. is the derivative of .

. 4 is just , and 8 forgets to divide by 3.

. Getting 0 usually means the antiderivative was written as instead of . The sine curve is above the axis on , so the answer must be positive.

Let , so : . Check by differentiating: . is the derivative, not the integral.

u-substitution: , . The integral becomes . The is used up by du, so it does not reappear in the answer.

Integration by parts with , : . Check: . differentiates to .

On , , so area = . is the area under alone, and adds the two areas instead of subtracting.

Fundamental Theorem of Calculus, Part 1: . You do not need an antiderivative, which is lucky, since has no elementary one. A factor would appear only if the upper limit were .

. 9 is the integral before dividing by the interval length, and 4.5 is the average of and .

Width . Left endpoints are 0 and 1: . The right sum is . For an increasing function, the left sum underestimates and the right sum overestimates the true value .

Disk method: . The solid is a cone with r = 2 and h = 2, and the cone formula agrees: . forgets the .

, so the antiderivative is . would need a in the numerator. goes with .

Displacement = . 10 m is , the velocity at t = 2, not a displacement.

. It converges because . The similar-looking diverges.

is an odd function, so the area below the axis on cancels the area above on : . A definite integral measures signed area, not total area.