7. Antiderivatives & Indefinite Integrals

Which of the following is an antiderivative of $f\left(x\right)=\cos (x^2-5)$?

Which of the following is an antiderivative of $f\left(x\right)=1+x^3$?

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

√x + 1/√x

Describe the set of antiderivatives of ƒ(x) = 1

Which of the following is an antiderivative of the function $f\left(x\right)=\cos x$?

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ e^{2x} / (4 + e^{2x}) dx

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / 2x³

Find the antiderivative of the following function.

$f\(\left\)(x\(\right\))=200$

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

p(x) = 3 sec² x

If $F\left(x\right)$ and $G\left(x\right)$ are both antiderivatives of $f\left(x\right)$, what must be true about $F\left(x\right)$ and $G\left(x\right)$?

If $f\left(x\right)$ and $g\left(x\right)$ are both antiderivatives of the same function, what must be true about $f\left(x\right)$ and $g\left(x\right)$?

Find the antiderivative of the following function.

$f\(\left\)(x\(\right\))=5x^4$

Which of the following is an antiderivative of $f\left(x\right)=tan(e^x+2)$ on $0\le x\le ln\left(2\right)$?

Applications

Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).

Find:

∫f(x) dx

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ (x²) / (4x³ + 7) dx