Find ∫_-2^2(x^5+x^3+x)/(x^4+x^2+1) dx+i^2026·log_2 1024.

bullet Factor the numerator: x^5+x^3+x=x(x^4+x^2+1) bullet The integrand simplifies to x, which is odd, so by symmetry: ∫_-2^2x dx=0 bullet Decode the second term: i^2026=i^2=-1 log_2 1024=10 bullet Combine: 0+(-1)(10)=-10

Daily Math Problems — June 16, 2026: Trigonometry, Integrals

dailymath

Problems & worked solutions

RO M1

Problem 1 — Solving Triangles

In $△ A B C$ , $A B = 5$ , $B C = 12$ , $A C = 13$ . The area equals:

Show answer & worked solution

A. $20$
B. $24$
C. $30$ ✓ correct
D. $60$

Since $5^{2} + 12^{2} = 13^{2}$ , $△ A B C$ is right-angled at $B$ . Area $= \frac{1}{2} \cdot 5 \cdot 12 = 30$ .

RO M1

Problem 2 — Integrals

Find $\int_{- 2}^{2} \frac{x^{5} + x^{3} + x}{x^{4} + x^{2} + 1} d x + i^{2026} \cdot \log_{2} 1024$ .

Show answer & worked solution

A. $0$
B. $10$
C. $- 1$
D. $- 10$ ✓ correct
E. $- 2^{2026}$
F. $Diverges$

$∙$ Factor the numerator:

$x^{5} + x^{3} + x = x (x^{4} + x^{2} + 1)$

$∙$ The integrand simplifies to $x$ , which is odd, so by symmetry:

$\int_{- 2}^{2} x d x = 0$

$∙$ Decode the second term:

$i^{2026} = i^{2} = - 1$

$\log_{2} 1024 = 10$

$∙$ Combine:

$0 + (- 1) (10) = - 10$

RO M1

Problem 3 — Trigonometric Equations

How many solutions does $2 \sin^{2} x - 3 \sin x + 1 = 0$ have on $[0, 2 π)$ ?

Show answer & worked solution

A. $2$
B. $3$ ✓ correct
C. $4$
D. $5$

$(2 \sin x - 1) (\sin x - 1) = 0$ . From $\sin x = \frac{1}{2}$ : $x = \frac{π}{6}, \frac{5 π}{6}$ . From $\sin x = 1$ : $x = \frac{π}{2}$ . Total: $3$ .

Problems & worked solutions

RO M1

Problem 1 — Solving Triangles

In $△ A B C$ , $A B = 5$ , $B C = 12$ , $A C = 13$ . The area equals:

Show answer & worked solution

A. $20$
B. $24$
C. $30$ ✓ correct
D. $60$

Since $5^{2} + 12^{2} = 13^{2}$ , $△ A B C$ is right-angled at $B$ . Area $= \frac{1}{2} \cdot 5 \cdot 12 = 30$ .

RO M1

Problem 2 — Integrals

Find $\int_{- 2}^{2} \frac{x^{5} + x^{3} + x}{x^{4} + x^{2} + 1} d x + i^{2026} \cdot \log_{2} 1024$ .

Show answer & worked solution

A. $0$
B. $10$
C. $- 1$
D. $- 10$ ✓ correct
E. $- 2^{2026}$
F. $Diverges$

$∙$ Factor the numerator:

$x^{5} + x^{3} + x = x (x^{4} + x^{2} + 1)$

$∙$ The integrand simplifies to $x$ , which is odd, so by symmetry:

$\int_{- 2}^{2} x d x = 0$

$∙$ Decode the second term:

$i^{2026} = i^{2} = - 1$

$\log_{2} 1024 = 10$

$∙$ Combine:

$0 + (- 1) (10) = - 10$

RO M1

Problem 3 — Trigonometric Equations

How many solutions does $2 \sin^{2} x - 3 \sin x + 1 = 0$ have on $[0, 2 π)$ ?

Show answer & worked solution

A. $2$
B. $3$ ✓ correct
C. $4$
D. $5$

$(2 \sin x - 1) (\sin x - 1) = 0$ . From $\sin x = \frac{1}{2}$ : $x = \frac{π}{6}, \frac{5 π}{6}$ . From $\sin x = 1$ : $x = \frac{π}{2}$ . Total: $3$ .