IXL - Evaluate definite integrals using substitution (Calculus practice)

Evaluate the integral. Write an exact, simplified answer.

∫

(

sin

)

(

cos

)

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question

Evaluate the integral. Write an exact, simplified answer.

∫

key idea

The substitution rule for definite integrals says that if

(

)

is a differentiable function whose range is an interval

and

(

)

is continuous on

then

∫

(

)

′

(

)

∫

(

)

(

)

(

)

solution

Use substitution to evaluate the definite integral.

The integrand contains the composite function

To simplify the integrand, try the substitution

Then

You can rearrange the differential to get

So,

was a good choice of substitution.

Substitute

and

into the integrand.

(

)

Then, find the limits of integration in terms of

To do so, substitute the original limits of integration, which are in terms of

into the equation for

When

(

)

When

(

)

Now, substitute the new limits of integration and integrate in terms of

∫

= 10

= 5

∫

= 4

= 3

1,024

–

243

781

Another method for evaluating the definite integral is to first find the indefinite integral in terms of

Use substitution to integrate in terms of

then rewrite the integral in terms of

∫

Integrate in terms of

Substitute

Now, evaluate the integral with the original limits of integration, which are in terms of

∫

= 10

= 5

(

)

–

(

)

1,024

–

243

781

$Confetti$

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