Applications of Differentiation VI LowerSixth Science Mathematics

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Test your knowledge on integration techniques and rules.

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Applications of Differentiation VI LowerSixth Science Mathematics


Applications of Differentiation VI LowerSixth Science MathematicsOnline version Test your knowledge on integration techniques and rules. by YAKILI LMS 1 The function e^x is its own derivative and integral. Yes No 2 integration by substitution requires finding du = f'(x) dx. Yes No 3 Tabular integration is a method for repeated integration by parts. Yes No 4 The derivative of arctan x is x/(1+x^2). Yes No 5 Partial fractions decomposition can be applied even if denominator has repeated factors without considering multiplicity. Yes No 6 The definite integral equals the sum of function values at endpoints. Yes No 7 The definite integral can be interpreted as accumulation. Yes No 8 Substitution (direct) uses a single u = g(x) to simplify integrand. Yes No 9 Long division is never needed in partial fractions. Yes No 10 The loop of integration by parts can be closed in finite steps for all functions. Yes No 11 A constant can be pulled out of a definite integral. Yes No 12 The definite integral from a to b depends on the path taken. Yes No 13 The integral of e^x from 0 to ∞ equals 0. Yes No 14 When using partial fractions, the denominator must factor into linear or irreducible quadratic factors. Yes No 15 The integral of 1/x from 0 to 1 diverges (is infinite). Yes No 16 The extension of integration by parts can handle product of more than two functions. Yes No 17 If f is continuous on [a,b], the fundamental theorem gives ∫_a^b f(x) dx = F(b) - F(a). Yes No 18 The formula for integration by parts can be applied recursively for products of more than two functions. Yes No 19 For rational functions, partial fractions always require factoring over real numbers; irreducible quadratics are allowed. Yes No 20 The method of substitution cannot handle integrals involving trigonometric functions. Yes No 21 Integration by parts formula: ∫u dv = uv - ∫v du. Yes No 22 The substitution method can yield integrals of the form ∫ f(g(x)) g'(x) dx. Yes No 23 If f is continuous on [a,b], the definite integral equals the limit of Riemann sums. Yes No 24 The antiderivative of sec^2 x is sec x. Yes No 25 Integration by parts is not useful for F(x) = x^2. Yes No 26 The definite integral from a to b of f(x) dx is independent of the path, unlike line integrals. Yes No 27 The definite integral of sin x from 0 to pi is 2. Yes No 28 Substitution by parts technique can be used to derive logarithmic integrals. Yes No 29 The area under a curve is always positive regardless of f. Yes No 30 The antiderivative of sec^2 x is tan x. Yes No 31 ∫ 1/x dx = ln\|x\| + C. Yes No 32 The substitution method cannot produce integrals of the form ∫ f(g(x)) g'(x) dx. Yes No 33 The substitution method can be used for integrals involving square roots. Yes No 34 Partial fractions decompose a rational function into simpler fractions. Yes No 35 The extension of integration by parts uses tabular method to simplify repeated dv/d u etc. Yes No 36 Boundary terms always vanish in definite integration by parts. Yes No 37 The definite integral value can be negative if the function is negative over [a,b]. Yes No 38 Substitution cannot be used for integrals involving square roots. Yes No 39 ∫_a^b f'(x) g(x) dx can be simplified using integration by parts. Yes No 40 The derivative of arctan x is 1/(1+x^2). Yes No 41 Partial fraction decomposition requires the numerator degree to be less than the denominator. Yes No 42 ∫ 1/x dx = x + C. Yes No 43 If deg numerator >= deg denominator, perform polynomial long division before partial fraction decomposition. Yes No 44 When integrating by parts, you must choose u and dv such that uv is easier than the original integrand. Yes No 45 Definite integral evaluates the net area under a curve between two limits. Yes No 46 The area between curves can be computed with definite integrals. Yes No 47 Boundary terms vanish in definite integration by parts if uv is zero at both bounds. Yes No 48 Partial fractions can simplify integrating rational functions. Yes No 49 Definite integrals always require the antiderivative evaluated at bounds. Yes No 50 Integration by substitution often involves choosing u and du. Yes No

Applications of Differentiation VI LowerSixth Science Mathematics

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Applications of Differentiation VI LowerSixth Science MathematicsOnline version

by YAKILI LMS

The function e^x is its own derivative and integral.

integration by substitution requires finding du = f'(x) dx.

Tabular integration is a method for repeated integration by parts.

The derivative of arctan x is x/(1+x^2).

Partial fractions decomposition can be applied even if denominator has repeated factors without considering multiplicity.

The definite integral equals the sum of function values at endpoints.

The definite integral can be interpreted as accumulation.

Substitution (direct) uses a single u = g(x) to simplify integrand.

Long division is never needed in partial fractions.

The loop of integration by parts can be closed in finite steps for all functions.

A constant can be pulled out of a definite integral.

The definite integral from a to b depends on the path taken.

The integral of e^x from 0 to ∞ equals 0.

When using partial fractions, the denominator must factor into linear or irreducible quadratic factors.

The integral of 1/x from 0 to 1 diverges (is infinite).

The extension of integration by parts can handle product of more than two functions.

If f is continuous on [a,b], the fundamental theorem gives ∫_a^b f(x) dx = F(b) - F(a).

The formula for integration by parts can be applied recursively for products of more than two functions.

For rational functions, partial fractions always require factoring over real numbers; irreducible quadratics are allowed.

The method of substitution cannot handle integrals involving trigonometric functions.

Integration by parts formula: ∫u dv = uv - ∫v du.

The substitution method can yield integrals of the form ∫ f(g(x)) g'(x) dx.

If f is continuous on [a,b], the definite integral equals the limit of Riemann sums.

The antiderivative of sec^2 x is sec x.

Integration by parts is not useful for F(x) = x^2.

The definite integral from a to b of f(x) dx is independent of the path, unlike line integrals.

The definite integral of sin x from 0 to pi is 2.

Substitution by parts technique can be used to derive logarithmic integrals.

The area under a curve is always positive regardless of f.

The antiderivative of sec^2 x is tan x.

∫ 1/x dx = ln|x| + C.

The substitution method cannot produce integrals of the form ∫ f(g(x)) g'(x) dx.

The substitution method can be used for integrals involving square roots.

Partial fractions decompose a rational function into simpler fractions.

The extension of integration by parts uses tabular method to simplify repeated dv/d u etc.

Boundary terms always vanish in definite integration by parts.

The definite integral value can be negative if the function is negative over [a,b].

Substitution cannot be used for integrals involving square roots.

∫_a^b f'(x) g(x) dx can be simplified using integration by parts.

The derivative of arctan x is 1/(1+x^2).

Partial fraction decomposition requires the numerator degree to be less than the denominator.

∫ 1/x dx = x + C.

If deg numerator >= deg denominator, perform polynomial long division before partial fraction decomposition.

When integrating by parts, you must choose u and dv such that uv is easier than the original integrand.

Definite integral evaluates the net area under a curve between two limits.

The area between curves can be computed with definite integrals.

Boundary terms vanish in definite integration by parts if uv is zero at both bounds.

Partial fractions can simplify integrating rational functions.

Definite integrals always require the antiderivative evaluated at bounds.

Integration by substitution often involves choosing u and du.