How do i do the anti-derivative?

Question

#dx/dy# sin(x/2)

Emma Aldridge · Accepted Answer

Assuming the question is #dy/dx=sin(x/2)#, the answer is #y=-2cos(x/2)# First, work out what trig function it's coming from. Sin differentiates to negative #cos#, so therefore negative #sin# differentiates to positive #cos#. Next divide by the differential of what's in the bracket. In the bracket: #(x/2)# #:.# derivative of that is just #(1/2)# #-cos(1/2x)# divided by a half #= -2cos(x/2)# which is your answer.

Elijah Abram · Answer

undefined To find the antiderivative (indefinite integral) of a function, you can use integration techniques such as:

1. **Power rule:** For functions in the form $ax^n$, where $a$ and $n$ are constants.
2. **Integration by parts:** Used for products of functions.
3. **Trigonometric integrals:** Integrating functions involving trigonometric functions like sine, cosine, tangent, etc.
4. **Substitution method:** Substituting a variable with another to simplify integration.
5. **Partial fractions:** Decomposing rational functions into simpler fractions.
6. **Trigonometric substitution:** Used for integrals involving radicals and trigonometric functions.
7. **Integration of rational functions:** Involves dividing the polynomial of the numerator by that of the denominator.
8. **Improper integrals:** Integrals with one or both bounds being infinite or integrals involving discontinuous functions.

Remember, the antiderivative is not unique as it includes an arbitrary constant of integration, typically denoted as $C$.