The anti-derivative of sqrt(x) :
sqrt(x)=x^(1/2)
The anti-derivative is x^(1/2+1) /(1/2+1) = (2/3) x^(3/2)
The anti-derivative is 4e^x is 4 e^x ( I hope you meant e to the power x)
The anti-derivative of -sin(x) is cos(x)
Adding, the anti-derivative is
(2/3) x^(3/2) + 4 e^x + cos(x) + C
Using u-substitution (where u = sinx), you'll find the antiderivative to be 0.5*sin2x + C.
(1-cosx)/sinx + sinx/(1- cosx) = [(1 - cosx)*(1 - cosx) + sinx*sinx]/[sinx*(1-cosx)] = [1 - 2cosx + cos2x + sin2x]/[sinx*(1-cosx)] = [2 - 2cosx]/[sinx*(1-cosx)] = [2*(1-cosx)]/[sinx*(1-cosx)] = 2/sinx = 2cosecx
The antiderivative of 9sinx is simply just -9cosx. It is negetive because the derivative of cosx should have been -sinx, however, the derivative provided is positive. Therefore, it means that there should be a negative with cosx in order to make that sinx postive. (negative times negative eguals positive)
d/dx(sinx-cosx)=cosx--sinx=cosx+sinx
-1
You will have to bear with the angle being represented by x because this browser will not allow characters from other alphabets!sin^2x + cos^2x = 1=> sin^2x = 1 - cos^x = (1 + cosx)(1 - cosx)Divide both sides by sinx (assuming that sinx is not zero).=> sinx = (1 + cosx)(1 - cosx)/sinxDivide both sides by (1 - cosx)=> sinx/(1 - cosx) = (1 + cosx)/sinx=> sinx/(1 - cosx) - (1 + cosx)/sinx = 0
2
(1 + tanx)/sinxMultiply by sinx/sinxsinx + tanxsinxDivide by sin2x (1/sin2x) = cscxcscx + tan(x)csc(x)tanx = sinx/cosx and cscx = 1/sinxcscx + (sinx/cosx)(1/sinx)sinx cancels outcscx + 1/cosx1/cosx = secxcscx + secx
If the first term is 7x^5, ∫7x^5 -cox dx is the expression. You can split this up into two integrals if that helps you visualize the terms. ∫7x^5dx - ∫cox dx. We know that the antiderivative of cosx is sinx, so that is our second term. In the first term, we must undo the power rule, adding one to the power and multiplying by the reciprocal of the power. This gives us (7/6)x^6. So, our final antiderivative expression is (7/6)x^6-sinx+C, with C being an arbitrary constant.
There is no sensible or useful simplification.
1/2(x-ln(sin(x)+cos(x)))
It is not possible to answer the question because it is ambiguous. The expression could be(x^2 - x + 1) / x - sinxor(x^2 - x + 1) / (x - sinx)Clearly, the answer will differ.This sort of ambiguity would have been avoided by using brackets.