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What is the indefinite integral?
An indefinite integral is a version of an integral that, unlike a definite integral, returns an expression instead of a number. The general form of a definite integral is: ∫ba f(x) dx. The general form of an indefinite integral is: ∫ f(x) dx. An example of a definite integral is: ∫20 x2 dx. An example of an indefinite integral is: ∫ x2 dx In the definite case, the answer is 23/3 - 03/3 = 8/3. In the indefinite case, the answer is x3/3 + C, where C is an arbitrary constant.
Find the indefinite integral x divided by x plus 1 quantity squared dx?
∫(x/(x+1)2)dx =∫((x+1-1)/(x+1)2)dx =∫(1/(x+1))dx - ∫(1/(x+1)2)dx u=x+1, du=dx ∫(1/u)du - ∫(1/u2)du =log(u) - (-1/u) + C =log(x+1) + 1/(x+1) + C
Derivative of 5ex plus 2?
5ex+2?d/dx(u+v)=du/dx+dv/dxd/dx(5ex+2)=d/dx(5ex)+d/dx(2)-The derivative of 5ex is:d/dx(cu)=c*du/dx where c is a constant.d/dx(5ex)=5*d/dx(ex)-The derivative of 2 is 0 because it is a constant.d/dx(5ex+2)=(5*d/dx(ex))+(0)d/dx(5ex+2)=5*d/dx(ex)-The derivative of ex is:d/dx(eu)=eu*d/dx(u)d/dx(ex)=ex*d/dx(x)d/dx(5ex+2)=5*(ex*d/dx(x))-The derivative of x is:d/dx(xn)=nxn-1d/dx(x)=1*x1-1d/dx(x)=1*x0d/dx(x)=1*(1)d/dx(x)=1d/dx(5ex+2)=5*(ex*1)d/dx(5ex+2)=5*(ex)d/dx(5ex+2)=5ex5ex+2?d/dx(cu)=c*du/dx where c is a constant.d/dx(5ex+2)=5*d/dx(ex+2)-The derivative of ex+2 is:d/dx(eu)=eu*d/dx(u)d/dx(ex+2)=ex+2*d/dx(x+2)d/dx(5ex+2)=5*(ex+2*d/dx(x+2))-The derivative of x+2 is:d/dx(u+v)=du/dx+dv/dxd/dx(x+2)=d/dx(x)+d/dx(2)d/dx(5ex+2)=5*[ex+2*(d/dx(x)+d/dx(2))]-The derivative of x is:d/dx(xn)=nxn-1d/dx(x)=1*x1-1d/dx(x)=1*x0d/dx(x)=1*(1)d/dx(x)=1-The derivative of 2 is 0 because it is a constant.d/dx(5ex+2)=5*[ex+2*(1+0)]d/dx(5ex+2)=5*[ex+2*(1)]d/dx(5ex+2)=5*[ex+2]d/dx(5ex+2)=5ex+2
What is the derivative of lnx raised to 4?
ln(x4)?d/dx(ln(u))=1/u*d/dx(u)d/dx(ln(x4))=[1/x4]*d/dx(x4)-The derivative of x4 is:d/dx(x4)=4x4-1d/dx(x4)=4x3d/dx(ln(x4))=[1/x4]*(4x3)d/dx(ln(x4))=4x3/x4d/dx(ln(x4))=4/x(lnx)4?Chain rule: d/dx(ux)=x(u)x-1*d/dx(u)d/dx(lnx)4=4(lnx)4-1*d/dx(lnx)d/dx(lnx)4=4(lnx)3*d/dx(lnx)-The derivative of lnx is:d/dx(ln(u))=1/u*d/dx(u)d/dx(lnx)=1/x*d/dx(x)d/dx(lnx)=1/x*(1)d/dx(lnx)=1/xd/dx(lnx)4=4(lnx)3*(1/x)d/dx(lnx)4=4(lnx)3/x
How do you find the derivative of 9 to the 5x?
95x?d/dx(au)=au*ln(a)*d/dx(u)d/dx(95x)=95x*ln(9)*d/dx(5x)-The derivative of 5x is:d/dx(cu)=c*du/dx where c is a constantd/dx(5x)=5*d/dx(x)d/dx(95x)=95x*ln(9)*(5*d/dx(x))-The derivative of x is:d/dx(x)=1x1-1d/dx(x)=1*x0d/dx(x)=1*(1)d/dx(x)=1d/dx(95x)=95x*ln(9)*(5*1)d/dx(95x)=95x*ln(9)*(5)-95x can simplify to (95)x, which equals 59049x.-ln(9) can simplify to ln(32), so you can take out the exponent to have 2ln(3).d/dx(95x)=59049x*2ln(3)*(5)d/dx(95x)=10*59049x*ln(3)
