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L3 Integration

Question Bank

Basic Integrals Reverse Chain, Logs & Division Trig Identities & Products Definite Integrals & Unknowns Areas Kinematics & Rates of Change dy/dx = f(x): Finding y Separable Differential Equations Growth, Decay & Cooling Trapezium & Simpson’s Rules Unfamiliar Contexts
Achieved

Given that

$ \int_1^5 g(x)\,dx = 9, $

find the value of

$ \int_5^1 \left(2g(x) - 3\right) dx. $

Achieved

Find the value of $k$, where $k > 4$, such that

$ \int_4^k 3\sqrt{x}\,dx = 38. $

You must use calculus and show the results of any integration needed to solve the problem.

Achieved

The graph of $y = f(x)$ below is symmetrical about the $y$-axis. The region between the graph and the $x$-axis from $x = -1$ to $x = 1$ has area $4.4$, and the region from $x = 1$ to $x = 3$ has area $7.6$.

Diagram

Find the value of

$ \int_0^3 f(x)\,dx. $

Achieved

Find the value of $k$ such that

$ \int_k^{25} \frac{5}{\sqrt{x}}\,dx = 30. $

You must use calculus and show the results of any integration needed to solve the problem.

Achieved

Find the value of $k$, where $k > 1$, such that

$ \int_1^k 4\sqrt[3]{x}\,dx = 45. $

You must use calculus and show the results of any integration needed to solve the problem.

Achieved

Given that

$ \int_1^4 f(x)\,dx = 7 $

and

$ \int_3^4 f(x)\,dx = 2, $

find the value of

$ \int_1^3 \left(f(x) + 2x\right) dx. $

Achieved

Find the value of the constant $k$ such that

$ \int_1^2 \left(\frac{k}{x^2} + 3\right) dx = 7. $

You must use calculus and show the results of any integration needed to solve the problem.

Achieved

Find, in terms of $k$,

$ \int_0^k 6e^{-3x}\,dx. $

Give your answer as simply as possible.

Merit

Find the value of $k$ such that

$ \int_0^k 2e^{0.5x}\,dx = 12. $

Give your answer in exact form.

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $k$ such that

$ \int_0^k 10e^{-2x}\,dx = 4. $

Give your answer in exact form.

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $k$ such that

$ \int_2^k \frac{6}{3x - 4}\,dx = 4. $

Give your answer in exact form.

You must use calculus and show the results of any integration needed to solve the problem.

Merit

The function $f(x) = a\sqrt{x} + b$, where $a$ and $b$ are constants, satisfies

$ \int_0^1 f(x)\,dx = 3 $

and

$ \int_0^4 f(x)\,dx = 28. $

Find the values of $a$ and $b$.

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find, in terms of $k$,

$ \int_0^k \frac{4e^{2x}}{e^{2x} + 3}\,dx. $

Give your answer as simply as possible.

Merit

Find the possible values of the constant $k$ such that

$ \int_0^3 (x - k)^2\,dx = 3. $

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $k$ such that

$ \int_2^k \frac{2x}{x^2 - 3}\,dx = \ln 6. $

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $k$ such that

$ \int_0^k \frac{2x + 9}{x + 3}\,dx = 2k + 6. $

Give your answer in exact form.

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of the constant $p$ such that

$ \int_p^{3p} (x - p)(3p - x)\,dx = 36. $

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $k$ such that

$ \int_0^{\ln k} \left(e^x + e^{-x}\right) dx = \frac{3}{2}. $

You must use calculus and show the results of any integration needed to solve the problem.

Merit

Find the value of $m$ such that

$ \int_m^{4m} \left(1 - \frac{6}{\sqrt{x}}\right) dx = 15. $

You must use calculus and show the results of any integration needed to solve the problem.

Merit

The constants $a$ and $k$ satisfy

$ \int_1^k 2x\,dx = a $

and

$ \int_1^4 \left(\frac{a}{\sqrt{x}} - 2k\right) dx = 18. $

Find the possible values of $a$ and $k$.

You must use calculus and show the results of any integration needed to solve the problem.