Integrals: Class 12 Maths Practice Questions

A civil engineer is designing a curved water channel whose cross-sectional area is modelled by the definite integral A = ∫₀^π x · sin x / (1 + cos²x) dx, where x is measured in metres. The engineer uses properties of definite integrals to evaluate this integral analytically.

A civil engineer is designing a curved water channel whose cross-sectional area is modelled by the definite integral

A = ∫₀^π x · sin x / (1 + cos²x) dx

where x is measured in metres.

(i) Using the property ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b−x) dx, show that A = (π/2) ∫₀^π sin x / (1 + cos²x) dx. [2]

(ii) Hence evaluate A and state the cross-sectional area of the channel in sq. metres. [2]

A civil engineer is designing a water channel whose cross-sectional area must be computed to determine water flow capacity. The cross-section is bounded above by the curve y = (x² + 4)/(x² + 3x + 2) along the horizontal axis from x = 0 to x = 1. To find the cross-sectional area, the engineer sets up the integral I = ∫₀¹ (x² + 4)/(x² + 3x + 2) dx.

A civil engineer is designing a water channel whose cross-sectional area must be computed to determine water flow capacity. The cross-section is bounded above by the curve y = (x² + 4)/(x² + 3x + 2) along the horizontal axis from x = 0 to x = 1.

To find the cross-sectional area, the engineer sets up the integral:

I = ∫₀¹ (x² + 4)/(x² + 3x + 2) dx

(i) Show that (x² + 4)/(x² + 3x + 2) can be written as 1 + A/(x+1) + B/(x+2) for some constants A and B. Find A and B. [1 mark]

(ii) Hence evaluate I = ∫₀¹ (x² + 4)/(x² + 3x + 2) dx and find the cross-sectional area of the channel. [3 marks]

A physicist studying wave interference models the intensity variation along a screen using the function I(x) = sin x · cos³x. The total energy contribution over a half-cycle [0, π/2] is E = ∫₀^(π/2) sin x · cos³x dx. A colleague proposes that J = ∫₀^(π/2) sin³x · cos x dx also equals E by King's property.

A physicist studying wave interference models the intensity variation along a screen using the function:

I(x) = sin x · cos³x

The total 'energy contribution' over a half-cycle [0, π/2] is computed as:

E = ∫₀^(π/2) sin x · cos³x dx

However, a colleague suggests that by symmetry, the same result can be obtained via the substitution u = cos x, and further proposes that a related integral:

J = ∫₀^(π/2) sin³x · cos x dx

also equals E, by applying the King's property (replacing x with π/2 − x).

Answer the following:
(i) Evaluate E = ∫₀^(π/2) sin x · cos³x dx using substitution. [1]
(ii) Evaluate J = ∫₀^(π/2) sin³x · cos x dx using substitution. [1]
(iii) Verify analytically that E = J by applying King's property to E, i.e., show that ∫₀^(π/2) sin x · cos³x dx = ∫₀^(π/2) sin³x · cos x dx, and hence find the value of E + J. [2]

A student is analysing the energy stored in a circuit, modelled by the integral I = ∫ (2x + 3)/((x + 1)(x² + 1)) dx. To evaluate this integral, the student attempts to decompose the integrand using partial fractions.

A student is analysing the energy stored in a circuit, modelled by the integral I = ∫ (2x + 3)/((x + 1)(x² + 1)) dx. To evaluate this integral, the student attempts to decompose the integrand using partial fractions.

(i) Write the partial fraction decomposition of (2x + 3)/((x + 1)(x² + 1)) in the standard form A/(x + 1) + (Bx + C)/(x² + 1). Find the values of A, B, and C. [1]

(ii) Hence evaluate I = ∫ (2x + 3)/((x + 1)(x² + 1)) dx. [3]

A civil engineer models the decorative arch of a bridge using the curve y = √(4 − x²) for x ∈ [−2, 2]. The area between this arch and the x-axis represents the inner decorative panel that needs to be painted. The engineer uses both exact integration and a numerical approximation to estimate this area.

A civil engineer is designing a curved arch for a bridge. The profile of the arch above the base is modelled by the curve y = √(4 − x²) for x ∈ [−2, 2]. To estimate the paint required for the arch's decorative inner panel, the engineer needs to compute the exact area enclosed between the arch and the x-axis.

(i) Identify the curve y = √(4 − x²) and state the limits of integration needed to find the enclosed area. [1]

(ii) Using the standard result ∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C, evaluate the definite integral to find the exact area of the panel. [2]

(iii) The engineer also considers a trapezoidal approximation using only the values at x = −2, 0, 2. Find the percentage error introduced by this approximation compared to the exact area (use π ≈ 3.14). [1]

A city engineer is designing a solar panel installation. The power output (in kilowatts) of a solar panel at time t hours after sunrise is modelled by the function P(t) = sin t + cos t, for 0 ≤ t ≤ π. The total energy output over a time interval [a, b] is given by ∫ₐᵇ P(t) dt kilowatt-hours.

A city engineer is designing a solar panel installation. The power output (in kilowatts) of a solar panel at time t hours after sunrise is modelled by the function P(t) = sin t + cos t, for 0 ≤ t ≤ π.

Based on this model, answer the following:
(i) Find the value of P(0) and P(π/2). [1]
(ii) Find the time t ∈ [0, π] at which P(t) is maximum, and state that maximum value. [1]
(iii) Find the total energy output (in kilowatt-hours) from t = 0 to t = π, i.e., evaluate ∫₀^π (sin t + cos t) dt. [2]

A civil engineer is designing a curved archway for a garden gate. The shape of the arch is modelled by the curve y = sin x + cos x for x ∈ [0, π/2]. To calculate the total area enclosed between the arch and the ground (x-axis), the engineer needs to evaluate a definite integral.

A civil engineer is designing a curved archway for a garden gate. The shape of the arch is modelled by the curve y = sin x + cos x for x ∈ [0, π/2]. To calculate the total area enclosed between the arch and the ground (x-axis), the engineer needs to evaluate a definite integral.

(i) Find the value of (sin x + cos x) at x = π/4.
(ii) Write the definite integral that represents the area under the arch from x = 0 to x = π/2.
(iii) Evaluate the integral to find the total area enclosed between the arch and the x-axis over [0, π/2].

OR

(iii) The engineer realises that the indefinite integral ∫(sin x + cos x) dx is needed for a related calculation. Evaluate ∫(sin x + cos x) dx.

A civil engineer is designing a water channel whose cross-sectional area (in m²) at a point x metres from one end is modelled by A(x) = ∫ [2x/(x² + 4) + 3/√(16 − x²)] dx, 0 ≤ x ≤ 4. The engineer uses standard integral results to find the general expression A(x) and then applies an initial condition to determine the channel's specific dimensions.

A civil engineer is designing a water channel whose cross-sectional area at a point x metres from one end is modelled by the function

A(x) = ∫ [2x / (x² + 4) + 3 / √(16 − x²)] dx, 0 ≤ x ≤ 4.

The engineer needs to find the general expression A(x) to plan the channel dimensions.

(i) Identify the standard integral form used to evaluate ∫ 2x/(x² + 4) dx and hence find its value. [1 mark]

(ii) Identify the standard integral form used to evaluate ∫ 3/√(16 − x²) dx and hence find its value. [1 mark]

(iii) Using parts (i) and (ii), write the complete expression for A(x). Given that A(0) = 2, find the particular value of the constant of integration and hence state the particular expression for A(x). [2 marks]

The value of ∫ eˣ (tan x + sec²x) dx is:

If ∫ (log x)/x dx = k(log x)² + C, then k is equal to:

If ∫ sec²(7 − 4x) dx = f(x) + C, then f(x) is :

∫ dx / (1 + cos 2x) is equal to :

Evaluate: ∫₀^π/2 dx / (1 + √tan x)

Find : ∫ 1/(√x · (1 + x)) dx

Find: ∫ dx / (x² + 4x + 8)

Find : ∫ (e^(3x) + 1) / e^(3x) dx

Find: ∫ (x² + 5) / [(x² + 1)(x² + 4)] dx

Find : ∫ 1 / [sin(x − a) sin(x − b)] dx

Find : ∫ (2x + 5) / (x² + 4x + 8) dx

Find ∫ e^(tan⁻¹x) · (1 + x + x²) / (1 + x²) dx.

Evaluate: ∫₁⁵ |x − 3| dx

Find : ∫ (x³ + x + 2) / ((x² + 1)(x + 1)) dx

A physics student is modelling the energy dissipated in a circuit. While computing the total energy, she needs to evaluate the integral

∫₀^(π/2) [x sin x cos x] / [sin⁴x + cos⁴x] dx

Using properties of definite integrals and appropriate substitutions, help the student evaluate this integral. Also state which property of definite integrals you used and why it simplifies the computation.

A civil engineer is designing a parabolic arch for a bridge. The vertical cross-section of the arch above the road level is modelled by the curve y = 4 − x², where y is the height (in metres) and x is the horizontal distance (in metres) from the centre of the arch. The road corresponds to the x-axis (y = 0).

A civil engineer is designing a parabolic arch for a bridge. The vertical cross-section of the arch above the road level is modelled by the curve y = 4 − x², where y is the height (in metres) and x is the horizontal distance (in metres) from the centre of the arch. The road corresponds to the x-axis (y = 0).

Based on this context, answer the following:

(i) Find the points where the arch meets the road level.

(ii) The engineer needs to check a symmetry property. Using the substitution x = −t, verify that
∫₋₂² (4 − x²) dx = 2 ∫₀² (4 − x²) dx.

(iii) Using the result from part (ii), evaluate the total cross-sectional area of the arch above the road, i.e., compute ∫₋₂² (4 − x²) dx.

OR

(iii) The engineer also considers the function f(x) = (4 − x²) + x³ over [−2, 2]. Using properties of definite integrals, evaluate ∫₋₂² [(4 − x²) + x³] dx.

A civil engineer is designing curved arches for a bridge. The cross-sectional areas under different arch profiles are modelled by definite integrals of polynomial functions over the interval [0, 2]. The engineer needs to evaluate these integrals accurately to ensure structural safety.

A civil engineer is designing a curved arch for a bridge. The cross-sectional area under the arch is modelled by the definite integral

∫₀² (3x² − 4x + 5) dx square metres.

(i) Using the power rule of integration, find the antiderivative F(x) of f(x) = 3x² − 4x + 5. [1 mark]
(ii) Hence evaluate the definite integral ∫₀² (3x² − 4x + 5) dx and state the cross-sectional area. [1 mark]
(iii) The engineer now considers a modified arch whose cross-section is given by

∫₀² |x² − 3x + 2| dx square metres.

Find the cross-sectional area of the modified arch. [2 marks]

A physicist models the intensity distribution of a laser beam cross-section along a radial axis using the function f(x) = x · e^(2x), where x (in cm) is the distance from the centre. The total intensity contribution from the centre to a point x = 1 cm is modelled by the integral I = ∫₀¹ x · e^(2x) dx.

A physicist models the intensity distribution of a laser beam cross-section along a radial axis using the function

f(x) = x · e^(2x)

where x (in cm) is the distance from the centre. The total intensity contribution from the centre to a point x = 1 cm is modelled by the integral

I = ∫₀¹ x · e^(2x) dx

(i) Identify the appropriate method to evaluate this integral and write down the ILATE choice. [1 mark]
(ii) Evaluate the indefinite integral ∫ x · e^(2x) dx. [1 mark]
(iii) Hence, evaluate I = ∫₀¹ x · e^(2x) dx and find its exact value. [2 marks]

A civil engineer is designing a curved arch for a bridge. The cross-sectional profile of the arch is modelled by the curve y = sin x + cos x for x ∈ [0, π/2], where x is measured in metres along the base and y is the height of the arch at position x. The engineer needs to compute the exact area under this curve to plan the quantity of filling material required.

A civil engineer is designing a curved arch for a bridge. The cross-sectional area under the arch is modelled by the function f(x) = sin x + cos x over the interval [0, π/2] (in metres). The engineer needs to compute the exact area under this curve to determine the material requirement.

(i) Write the definite integral that represents the area under the arch from x = 0 to x = π/2.
(ii) Evaluate the integral ∫₀^(π/2) sin x dx.
(iii) Hence, find the total area under the arch, i.e., evaluate ∫₀^(π/2) (sin x + cos x) dx. Also verify that your answer is consistent with the physical requirement that the area must be positive.

A civil engineer is designing a parabolic arch bridge modelled by the curve y = 4 − x², where the road deck lies along the x-axis. The arch spans from x = −2 to x = 2 (where the curve meets the x-axis). During structural analysis, various integrals arise in computing cross-sectional areas, centroids, and stress distributions modelled by Fourier-type expressions.

A civil engineer is designing a parabolic arch bridge. The arch is modelled by the curve y = 4 − x², and the road deck runs along the x-axis. During a load-stress analysis, the engineer needs to evaluate the following two integrals:

(i) Find the total cross-sectional area (in sq. units) enclosed between the arch y = 4 − x² and the road deck (x-axis).

(ii) To compute the centroid height of the arch, the engineer must evaluate:
∫₋₂² (4 − x²) dx / (Total area found in part (i))

However, before that, evaluate the definite integral:
∫₀^(π/2) x · sin x dx
using integration by parts, which appears in the Fourier stress model for the bridge.

(iii) Using the property of definite integrals, evaluate:
∫₀^π x · sin x / (1 + cos²x) dx

A civil engineer is designing a decorative arch for a park entrance. The arch is modelled by the curve y = √(6 + 4x − x²) above the x-axis. The engineer needs to calculate the exact area enclosed between the arch and the x-axis to determine the amount of material required.

A civil engineer is designing a decorative arch for a park entrance. The arch is modelled by the curve y = √(6 + 4x − x²) above the x-axis. The engineer needs to calculate the exact area (in square units) enclosed between the arch and the x-axis to determine the amount of material required.

(i) Express 6 + 4x − x² in the form a² − (x − b)² and identify the values of a and b. [1]
(ii) Write down the limits of integration (the x-intercepts of the arch) and set up the definite integral for the required area. [1]
(iii) Hence evaluate the integral and find the exact area enclosed between the arch and the x-axis. [2]

A civil engineer is designing a curved garden wall whose cross-sectional area (in square metres) is modelled by the definite integral

∫₀¹ x · tan⁻¹x dx

Using integration by parts, find the exact cross-sectional area. Also verify that the result lies between 1/4 and 1/2 (i.e., 1/4 < Area < 1/2), justifying your reasoning.