Lesson test: Techniques of integration

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Techniques of integration

1.[3p]
Evaluate $\int_{0}^{2} x e^{x^{2}} d x$ , to two decimal places.

CorrectNot quite: 26.80

2.[3p]
Evaluate $\int_{0}^{π / 2} \sin^{3} x \cos x d x$ .

CorrectNot quite: 0.25

3.[3p]
Evaluate $\int_{1}^{e} \ln x d x$ .

CorrectNot quite: 1

4.[2p]
In integration by parts, how should the split be chosen?

Let $u$ be the factor that gets simpler when differentiated and $d v$ the factor you can integrate

Correct

Let $u$ always be the first factor written

The answer is: Let $u$ be the factor that gets simpler when differentiated and $d v$ the factor you can integrate

Let $d v$ be whichever factor is a polynomial

The answer is: Let $u$ be the factor that gets simpler when differentiated and $d v$ the factor you can integrate

The choice never matters, since both give the same work

The answer is: Let $u$ be the factor that gets simpler when differentiated and $d v$ the factor you can integrate

5.[3p]
The function $e^{- x^{2}}$ has no antiderivative.

True

The answer is: False

False

Correct

6.[3p]
What did Liouville prove about integrals such as $\int e^{- x^{2}} d x$ ?

That no elementary antiderivative exists, so failing to find one is not a failure of skill

Correct

That the integral diverges

The answer is: That no elementary antiderivative exists, so failing to find one is not a failure of skill

That the integrand is not continuous

The answer is: That no elementary antiderivative exists, so failing to find one is not a failure of skill

That only numerical methods converge for it

The answer is: That no elementary antiderivative exists, so failing to find one is not a failure of skill

7.[3p]
Estimate $\int_{0}^{1} e^{- x^{2}} d x$ with Simpson's rule on four strips, to four decimal places.

CorrectNot quite: 0.7469

8.[2p]
European mains is quoted as 230 V RMS. What is the peak voltage, to the nearest volt?

CorrectNot quite: 325

9.[3p]
Match each technique to the differentiation rule it inverts or the structure it handles.

Substitution
Integration by parts
Partial fractions
Simpson's rule

the chain rule
the product rule
no antiderivative available
a quotient of polynomials

Show the answer

Substitution: the chain rule Integration by parts: the product rule Partial fractions: a quotient of polynomials Simpson's rule: no antiderivative available

1.[3p] Evaluate ∫02xex2dx, to two decimal places.

2.[3p] Evaluate ∫0π/2sin3xcosxdx.

3.[3p] Evaluate ∫1elnxdx.

4.[2p] In integration by parts, how should the split be chosen?

5.[3p] The function e−x2 has no antiderivative.

6.[3p] What did Liouville prove about integrals such as ∫e−x2dx?

7.[3p] Estimate ∫01e−x2dx with Simpson's rule on four strips, to four decimal places.

8.[2p] European mains is quoted as 230 V RMS. What is the peak voltage, to the nearest volt?

9.[3p] Match each technique to the differentiation rule it inverts or the structure it handles.

1.[3p]
Evaluate $\int_{0}^{2} x e^{x^{2}} d x$ , to two decimal places.

2.[3p]
Evaluate $\int_{0}^{π / 2} \sin^{3} x \cos x d x$ .

3.[3p]
Evaluate $\int_{1}^{e} \ln x d x$ .

4.[2p]
In integration by parts, how should the split be chosen?

5.[3p]
The function $e^{- x^{2}}$ has no antiderivative.

6.[3p]
What did Liouville prove about integrals such as $\int e^{- x^{2}} d x$ ?

7.[3p]
Estimate $\int_{0}^{1} e^{- x^{2}} d x$ with Simpson's rule on four strips, to four decimal places.

8.[2p]
European mains is quoted as 230 V RMS. What is the peak voltage, to the nearest volt?

9.[3p]
Match each technique to the differentiation rule it inverts or the structure it handles.