9.1.3 Antiderivatives of Trigonometric and Exponential Functions

Question 1

Antiderivatives of Trigonometric and Exponential Functions

Answer 1

• Given two functions, f and F, F is an antiderivative of f if F ′ (x ) = f(x ). Antidifferentiation is a process or operation that reverses differentiation.
• Discover integration formulas by looking at differentiation formulas backwards.

Question 2

note

Answer 2

• Here are some antiderivative formulas.
• Notice that some functions that are easy to
  differentiate are not as easy to integrate. It is
  generally the case that it is easier to differentiate
  than integrate.
• To evaluate this indefinite integral, start by applying
  the sum rule.
• Now you can evaluate the integral of each term
  individually.
• Remember, when using the power rule for
  integration, you must multiply by the reciprocal of
  the new exponent.
• You can always check that your answer is correct
  by taking the derivative.

Question 3

Find f(x) so that f′(x)=−4e^x−6sinx.

Answer 3

−4e^ x + 6 cos x + C

Question 4

Evaluate the integral. ∫sec^2xdx

Answer 4

tanx + C

Question 5

Evaluate the integral. ∫sinx dx

Answer 5

− cos x + C

Question 6

Evaluate the integral: ∫(2sinx+3cosx) dx.

Answer 6

−2 cos x + 3 sin x + C

Question 7

Evaluate the integral ∫sin2x/cosx dx.

Answer 7

−2 cos x + C

Question 8

Evaluate: ∫secx(tanx+secx) dx.

Answer 8

sec x + tan x + C

Question 9

Evaluate the integral: ∫(1+sin^2θcscθ) dθ

Answer 9

θ−cosθ+C

Question 10

Evaluate the integral:∫3exdx.

Answer 10

3e^x+C

Question 11

Evaluate:

∫tan^2xdx

Answer 11

tan x − x + C