Trig calculus

Author: SolveSheep

Integrals, derivatives, limits and sums that involve trigonometric functions from easy things to more difficult ones.




Not logged in? Click here to log in.
1.0 Calculus
Definite Integral of \(\frac{1}{\sqrt{1-x^2}}\) from 0 to 1

Compute the definite integral \( \int_0^1 \frac{1}{\sqrt{1-x^2}} \, dx \).
1.0 Calculus
Definite Integral of \(\sec^2 x\)

Evaluate the definite integral \( \int_0^{\pi/4} \sec^2 x \, dx \).
1.0 Calculus
Derivative of \(f(x) = \frac{\sin x}{x}\)

Find the derivative of the function \(f(x) = \frac{\sin x}{x}\) with respect to \(x\).
1.0 Calculus
Derivative of \(f(x) = \ln(\tan x)\)

Find the derivative of the function \(f(x) = \ln(\tan x)\) with respect to \(x\).
1.0 Calculus
Derivative of \(f(x) = x \sin x\)

Find the derivative of the function \(f(x) = x \sin x\).
1.0 Calculus
Limit of \(\frac{\sin(3x)}{\tan(5x)}\) as \(x \to 0\)

Calculate the limit \( \lim_{x \to 0} \frac{\sin(3x)}{\tan(5x)} \).
1.5 Calculus Trigonometry
Definite Integral of \(\cos^2 x\)

Evaluate the definite integral:$$ \int_{0}^{\pi/2} \cos^2 x \, dx $$
1.5 Calculus
Definite Integral of \(\sin^2 x\)

Compute the definite integral \( \int_0^{\pi/2} \sin^2 x \, dx \).
1.5 Calculus
Derivative of \(h(x) = \arcsin(\sqrt{x})\)

Find the derivative of the function \(h(x) = \arcsin(\sqrt{x})\) with respect to \(x\).
1.5 Calculus
Indefinite Integral of \(\frac{\cos x}{1 + \sin^2 x}\)

Find the indefinite integral \( \int \frac{\cos x}{1 + \sin^2 x} \, dx \).
1.5 Calculus
Indefinite Integral of \(\tan(2x)\)

Evaluate the indefinite integral \( \int \tan(2x) \, dx \).
1.5 Calculus
Limit of \(\frac{1 - \cos x}{x^2}\) as \(x \to 0\)

Compute the limit \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} \).
1.5 Calculus
Limit of \(\frac{\cos x}{\pi/2 - x}\) as \(x \to \pi/2\)

Calculate the limit \( \lim_{x \to \pi/2} \frac{\cos x}{\pi/2 - x} \).
1.5 Calculus
Limit of \(x \sin\left(\frac{1}{x}\right)\) as \(x \to \infty\)

Evaluate the limit \( \lim_{x \to \infty} x \sin\left(\frac{1}{x}\right) \).
2.0 Calculus
Indefinite Integral of \(\cos^3 x\)

Evaluate the indefinite integral \( \int \cos^3 x \, dx \).
2.0 Calculus
Indefinite Integral of \(\frac{1}{\sin x \cos x}\)

Evaluate the indefinite integral \( \int \frac{1}{\sin x \cos x} \, dx \).
2.0 Calculus
Indefinite Integral of \(e^x \cos x\)

Calculate the indefinite integral \( \int e^x \cos x \, dx \).
2.0 Calculus
Limit of \(\frac{\sin x - x}{x^3}\) as \(x \to 0\)

Evaluate the limit:$$ \lim_{x \to 0} \frac{\sin x - x}{x^3} $$
2.0 Calculus
Second Derivative of \(g(x) = \cos(x^2)\)

Determine the second derivative of the function \(g(x) = \cos(x^2)\) with respect to \(x\).
2.0 Algebra Calculus
Sum of Maclaurin Series for Cosine

Find the sum of the infinite series \( \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{(2n)!} \).
2.5 Calculus
Derivative of \(y = (\sin x)^{\cos x}\)

Differentiate the function \(y = (\sin x)^{\cos x}\) with respect to \(x\).
2.5 Algebra Trigonometry
Finite Sum of \(\sin(kx)\)

Find a closed-form expression for the sum \( \sum_{k=1}^{N} \sin(kx) \).
2.5 Calculus Trigonometry
Indefinite Integral of \(\sec^3 x\)

Find the indefinite integral:$$ \int \sec^3 x \, dx $$
2.5 Calculus
Limit of \((\sin x)^x\) as \(x \to 0^+\)

Determine the limit \( \lim_{x \to 0^+} (\sin x)^x \).
3.0 Algebra Calculus Trigonometry
Sum of Telescoping Series with Arctangent

Find the sum of the series:$$ \sum_{n=1}^{N} \arctan\left(\frac{1}{n^2+n+1}\right) $$