(a) Find the derivatives of sin(x2 +1) and sin(x3+1). (b) Use your answer to part (a) to

(a) Find the derivatives of sin(x2 +1) and sin(x3+1). (b) Use your answer to part (a) to find antiderivatives of: (i) x cos(x2 + 1) (ii) x2 cos(x3 + 1) (c) Find the general antiderivatives of: (i) x sin(x2 + 1) (ii) x2 sin(x3 +1)

In Problems 25 explain how you can tell if substitution can be used to find an antiderivative. < x(1 5x2)5 dx

In Problems 25 explain how you can tell if substitution can be used to find an antiderivative. < ln x x dx

In Problems 25 explain how you can tell if substitution can be used to find an antiderivative. < x ln x dx

In Problems 25 explain how you can tell if substitution can be used to find an antiderivative. < sin9 t cos t dt

Find the integrals in Problems 647. Check your answers by differentiation. < 2x(x2 + 1)5dx

Find the integrals in Problems 647. Check your answers by differentiation. < x x2 + 4 dx

Find the integrals in Problems 647. Check your answers by differentiation. < (5x 7)10dx

Find the integrals in Problems 647. Check your answers by differentiation. < x4x2 + 1dx

Find the integrals in Problems 647. Check your answers by differentiation. < 2qeq2+1dq

Find the integrals in Problems 647. Check your answers by differentiation. < 5e5t+2dt

Find the integrals in Problems 647. Check your answers by differentiation. < xex2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < 100e0.2tdt

Find the integrals in Problems 647. Check your answers by differentiation. < t2(t3 3)10 dt

Find the integrals in Problems 647. Check your answers by differentiation. < x sin(x2)dx

Find the integrals in Problems 647. Check your answers by differentiation. < x(x2 4)7/2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < x(x2 + 3)2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < 1 (3x + 1)2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < 4x3 x4 + 1 dx

Find the integrals in Problems 647. Check your answers by differentiation. < 12x2 cos(x3)dx

Find the integrals in Problems 647. Check your answers by differentiation. < (2t 7)73 dt

Find the integrals in Problems 647. Check your answers by differentiation. < (x2 +3)2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < y2(1 + y)2 dy

Find the integrals in Problems 647. Check your answers by differentiation. < sin (cos + 5)7 d

Find the integrals in Problems 647. Check your answers by differentiation. < sin6 cos d

Find the integrals in Problems 647. Check your answers by differentiation. < cos 3t sin 3t dt

Find the integrals in Problems 647. Check your answers by differentiation. < sin(3 t) dt

Find the integrals in Problems 647. Check your answers by differentiation. < t 1 + 3t2 dt

Find the integrals in Problems 647. Check your answers by differentiation. < x2ex3+1 dx

Find the integrals in Problems 647. Check your answers by differentiation. < sin3 cos d

Find the integrals in Problems 647. Check your answers by differentiation. < x sin(4x2)dx

Find the integrals in Problems 647. Check your answers by differentiation. < sin2 x cos xdx

Find the integrals in Problems 647. Check your answers by differentiation. < e3x4dx

Find the integrals in Problems 647. Check your answers by differentiation. < 1 + ex x + ex dx

Find the integrals in Problems 647. Check your answers by differentiation. < xe3x2 dx

Find the integrals in Problems 647. Check your answers by differentiation. < x43x2 + 4dx

Find the integrals in Problems 647. Check your answers by differentiation. < ex ex ex + ex dx

Find the integrals in Problems 647. Check your answers by differentiation. < (ln z)2 z dz

Find the integrals in Problems 647. Check your answers by differentiation. < y y2 + 4 dy

Find the integrals in Problems 647. Check your answers by differentiation. < q 5q2 + 8 dq

Find the integrals in Problems 647. Check your answers by differentiation. < et + 1 et + t dt

Find the integrals in Problems 647. Check your answers by differentiation. < e y y dy

Find the integrals in Problems 647. Check your answers by differentiation. < cosx x dx

Find the integrals in Problems 647. Check your answers by differentiation. < x + 1 x2 + 2x + 19 dx

Find the integrals in Problems 647. Check your answers by differentiation. < et et + 1 dt

Find the integrals in Problems 647. Check your answers by differentiation. < sin6(5) cos(5) d

Find the integrals in Problems 647. Check your answers by differentiation. < x cos(x2) 4sin(x2) dx

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) < x sin(x2) dx (b) < x2 sinxdx (c) < x2 1 + x2 dx (d) < x (1 + x2)2 dx (e) < x3ex2 dx (f) < sin x 2 + cos x dx

Use substitution to express each of the following integrals as a multiple of =b a (1/w) dw for some a and b. Then evaluate the integrals. (a) < 1 0 x 1 + x2 dx (b) < /4 0 sin x cos x dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 2 0 x(x2 + 1)2 dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 3 0 2x x2 +1 dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < /2 0 ecos sin d

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 4 1 e x x dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 3 0 1 t+ 1 dt

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < e2 1 1 t +2 dt

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 2 0 x (1 + x2)2 dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 1 0 2tet2 dt

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 2 1 x +2dx

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems 5059. < 3 1 dt (t+ 7)2

Under f(x) = xex2 between x = 0 and x = 2.

Under f(x) = 1/(x + 1) between x = 0 and x = 2.

Find the exact average value of f(x) = 1/(x + 1) on the interval x = 0 to x = 2. Sketch a graph showing the function and the average value.

Suppose =2 0 g(t) dt = 5. Calculate the following: (a) < 4 0 g(t/2) dt (b) < 2 0 g(2 t) dt

(a) Find =(x+ 5)2 dx in two ways: (i) By multiplying out (ii) By substituting w = x+ 5 (b) Are the results the same? Explain.

Find = 4x(x2 + 1) dx using two methods: (a) Do the multiplication first, and then antidifferentiate. (b) Use the substitution w = x2 + 1. (c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

(a) Find = sin cos d. (b) You probably solved part (a) by making the substitution w = sin or w = cos. (If not, go back and do it that way.) Now find = sin cos d by making the other substitution. (c) There is yet another way of finding this integral which involves the trigonometric identities sin(2) = 2sin cos cos(2) = cos2 sin2 . Find = sin cos d using one of these identities and then the substitution w = 2. (d) You should now have three different expressions for the indefinite integral = sin cos d. Are they really different? Are they all correct? Explain.