Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{2}{x^{2} + 4}\), on real x.

Find K in the primitive \(\displaystyle K\arctan (\frac{x}{2})+ C\).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{2}{x^{2} + 1}\), on real x.

Find K in the primitive \(\displaystyle K\arctan (\frac{x}{1})+ C\).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{3}{x^{2} + 9}\), on real x.

Find f(0).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{4}{x^{2} + 16}\), on real x.

Find f(0).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{4}{x^{2} + 16}\), on real x.

Find the integral from 0 to 4, divided by π.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{2}{x^{2} + 1}\), on real x.

Find the integral from 0 to 1, divided by π.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{5}{x^{2} + 25}\), on real x.

Find the integral from 0 to 10.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{2}{x^{2} + 1}\), on real x.

Find the integral from 0 to 2.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(f(x)=3 x^{2} e^{3 x}\), on real x.

After factoring \(\displaystyle e^{3x}\) from a primitive, find the coefficient of \(\displaystyle x^2\).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(f(x)=4 x e^{x}\), on real x.

After factoring \(\displaystyle e^{1x}\) from a primitive, find the coefficient of \(\displaystyle x^1\).

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(f(x)=4 x \left(x^{2} + 3\right)\), on real x.

Use the substitution \(u=x^2+3\).

Find the definite integral from 0 to 1.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{5 x}{\sqrt{x^{2} + 4}}\), on real x.

Use the substitution \(u=x^2+4\).

Find the definite integral from 0 to 1.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{10 x + 20}{x^{2} + 4 x + 17}\), on real x.

Find the definite integral from 0 to 1.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The integrand is \(\displaystyle f(x)=\frac{4 x + 2}{x^{2} + x + 2}\), on real x.

Find the definite integral from 0 to 1.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The curve is \(y=\left(3 - x\right) \left(x - 1\right)\). Consider the finite region bounded by the curve and the x-axis.

Find the maximum height above the x-axis.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.

Maths4U Original · Short practice

The curve is \(y=\left(5 - x\right) \left(2 x - 4\right)\). Consider the finite region bounded by the curve and the x-axis.

Find the maximum height above the x-axis.

Enter one number or fraction. Keep exact values during your working; use at least six decimal places for an approximate final answer. A calculator may be used.