Question : Evaluate the integral using integration by parts. 21) integral(x)sqrt(x + 1)dx : 2163446

Evaluate the integral using integration by parts.

21) ∫(x)√(x + 1)dx

A) (2/3)x(x + 1)^3/2 - (4/15)(x + 1)^5/2 + C

B) x(x + 1)^3/2 - (x + 1)^5/2 + C

C) (2/3)(x + 1)^3/2 - (4/15)(x + 1)^5/2 + C

D) (x + 1)^3/2 + (x + 1)^5/2 + C

22) ∫(e^axcosbxdx)

A) e^ax(sinbx + cosbx) + C

B) e^ax(sinbx - cosbx) + C

C) e^ax((bsinbx + acosbx/a² + b²)) + C

D) (1/2b)e^axcos²bx + C

23) ∫(e^axsinbxdx)

A) (1/2b)e^axcos²bx + C

B) e^ax(sinbx + cosbx) + C

C) e^ax((asinbx - bcosbx/a² + b²)) + C

D) e^ax(sinbx - cosbx) + C

24) ∫(e^2xsin3xdx)

A) - e^2x(sin3x - 3cos3x) + C

B) (1/13)e^2x(sin3x - 3cos3x) + C

C) (1/13)e^2x(2sin3x - 3cos3x) + C

D) e^2x(sin3x - cos3x) + C

25) ∫(sec⁴xdx)

A) (1/5)sec⁵xtanx + C

B) 4sec³xtanx+ C

C) (1/3)sec²xtanx + (2/3)tanx + C

D) sec²xtanx + tanx + C

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

26) Does this integral ∫(2xe^xdx) = 2e^x(x - 1) + C?

Enter "yes" or "no".

27) Does this integral ∫(t³cost dt) = (t³ - 6t)sint + 3(t² - 2)cost + C?

Enter "yes" or "no".

28) Does this integral ∫((ln(2x)/x²)dx) = - (1 + ln2 + lnx/x) + C?

Enter "yes" or "no".

29) Does this integral ∫(3x√(4x + 3)dx) = (3(2x - 1)(4x + 3)^3/2/20) + C?

Enter "yes" or "no".

30) Does this integral ∫(x²sin2xdx) = ((1 - 2x²)cos²x/4) + (xsin2x/2) + C?

Enter "yes" or "no".

31) Does this integral ∫(xe^xsinxdx) = (-e^x((x - 1)cosx +xsinx)/2) + C?

Enter "yes" or "no".

32) Does this integral ∫(xsin10xdx) = (1/100)sin10x - (x/10)cos10x + C?

Enter "yes" or "no".

33) Does this integral ∫(xcos5xdx) = (1/5)xsin5x - (1/25) cos5x + C?

Enter "yes" or "no".

34) Does this integral ∫(x³lnxdx) = (1/4)x⁴lnx + (1/16)x⁴ + C?

Enter "yes" or "no".

35) Does this integral ∫(xe^-3xdx) = - (1/3)xe^-3x - (1/9)e^-3x + C?

Enter "yes" or "no".

36) Does this integral ∫(xsec²xdx) = xtanx + ln|cosx| + C?

Enter "yes" or "no".

37) Does this integral ∫(cosx/(3sinx + 1)²)dx = (1/3(3sinx + 1)) + C?

Enter "yes" or "no".

38) Does this integral ∫(x²e^xdx) = x²e^x - 2(xe^x - e^x) + C?

Enter "yes" or "no".