Trigonometric Integral Solver

Math Calculus • Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

7. Trigonometric Integral Solver

Solves common trig-power integrals step-by-step and supports definite bounds with shaded area. Includes pan/zoom graph, constant multipliers, and separate arguments \(k_1x\), \(k_2x\).

Inputs

Integral family

Product families support different coefficients \(k_1,k_2\). Exact symbolic methods are used when \(k_1=k_2\); otherwise numeric-only.

Argument coefficient \(k_1\)

Supports pi, e, + - * / ^ and parentheses. (nonzero)

Argument coefficient \(k_2\)

Shown only for product families.

Coefficient

Multiplier for the first trig factor.

Power \(m\)

Nonnegative integer (recommended \(\le 20\)).

Coefficient

Multiplier for the second trig factor.

Power \(n\)

Nonnegative integer (recommended \(\le 20\)).

Method

Auto chooses a standard strategy. For product families with \(k_1\ne k_2\), the solver switches to numeric-only.

Integral mode

Definite mode shades the region on the graph and computes the value (exact when possible, otherwise numeric).

Period shortcuts

good for periodic cases

Options

Show steps Show \(f(x)\) Show \(F(x)+C\) (if available) Grid Shade \([a,b]\) (definite) Show Weierstrass (when applicable)

Presets

Click a preset to load and evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom. In definite mode, shaded region shows signed area between \(a\) and \(b\).

Center \(c\)

Half-width \(w\)

Legend

\(f(x)\) \(F(x)\) (if shown) shaded \([a,b]\)

x: 0, y: 0, zoom(px/unit): 80

Result

Choose a family, coefficients, powers, and \(k_1,k_2\), then click Evaluate.

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7. Trigonometric Integral Solver — Theory

This tool solves many common integrals of trig powers, and supports definite bounds with a shaded area. For product families, the arguments may use separate coefficients:

\[ \int \sin^m(k_1x)\cos^n(k_2x)\,dx,\quad \int \tan^m(k_1x)\sec^n(k_2x)\,dx,\quad \int \cot^m(k_1x)\csc^n(k_2x)\,dx. \]

The standard symbolic techniques (odd/even splitting, substitutions, power-reduction) require the trig factors to share the same inside angle, so the calculator gives exact symbolic output when \(k_1=k_2\). If \(k_1\neq k_2\) in a product family, the tool switches to a numeric evaluation (definite mode) while still graphing and shading correctly.

1) Handling coefficients in the argument

\[ X = kx \quad\Rightarrow\quad dX = k\,dx,\quad dx=\frac{dX}{k}. \]

\[ \int f(kx)\,dx = \frac{1}{k}\int f(X)\,dX. \]

This is why, when \(k_1=k_2=k\), the tool can reduce everything to a single variable \(X=kx\) and then apply standard methods.

2) \(\sin^m x\cos^n x\) (odd/even rules)

If \(m\) is odd: save one \(\sin x\) and convert the rest using \(\sin^2x=1-\cos^2x\), then use \(u=\cos x\).
If \(n\) is odd: save one \(\cos x\) and convert the rest using \(\cos^2x=1-\sin^2x\), then use \(u=\sin x\).
If both are even: use power-reduction identities \(\sin^2x=\frac{1-\cos 2x}{2}\), \(\cos^2x=\frac{1+\cos 2x}{2}\), then integrate a sum of sines/cosines.

3) \(\tan^m x\sec^n x\) (standard strategies)

If \(n\) is even (and \(\ge 2\)): save \(\sec^2x\,dx\), rewrite the rest using \(\sec^2x=1+\tan^2x\), then \(u=\tan x\).
If \(m\) is odd (and \(n\ge 1\)): save \(\sec x\tan x\,dx\), rewrite the rest using \(\tan^2x=\sec^2x-1\), then \(u=\sec x\).

4) Single-function powers

The tool includes single-function families (no need to set the other power to 0), e.g. \(\int \tan^m(k_1x)\,dx\), \(\int \sec^m(k_1x)\,dx\), \(\int \cot^m(k_1x)\,dx\), \(\int \csc^m(k_1x)\,dx\). It uses standard reduction formulas for \(\tan^m\), \(\cot^m\), \(\sec^m\), \(\csc^m\).

5) Definite integrals and shading

In definite mode, the calculator shades the region between \(a\) and \(b\) and computes \(\int_a^b f(x)\,dx\). If a closed form \(F\) is available, it uses \(F(b)-F(a)\); otherwise it uses a numerical method.

Frequently Asked Questions

What kinds of trigonometric integrals can this solver handle?

It targets common trig-power families such as sin^m(k1x)cos^n(k2x), tan^m(k1x)sec^n(k2x), and cot^m(k1x)csc^n(k2x), plus single-function power cases like integral sin^m(k1x) dx or integral sec^m(k1x) dx.

Why does the solver switch to numeric-only when k1 is not equal to k2 in a product family?

The standard symbolic strategies rely on both trig factors sharing the same inside angle so substitutions and power-reduction identities apply cleanly. When k1 != k2, the tool computes the definite value numerically while still graphing and shading the interval correctly.

How are sin^m(x)cos^n(x) integrals typically solved?

If one power is odd, one factor is saved and the rest is rewritten using sin^2(x)=1-cos^2(x) or cos^2(x)=1-sin^2(x) and a substitution u=cos(x) or u=sin(x) is used. If both powers are even, power-reduction identities convert the integrand into sums of sines and cosines with multiple angles.

What does the definite-mode shaded region represent?

It shows the signed area under f(x) between a and b. Regions above the x-axis contribute positively and regions below contribute negatively to the integral value.

What is the Weierstrass option used for?

When applicable, it displays an alternative substitution-based form that can help rewrite trigonometric expressions into a rational form for integration. It is presented as an optional view alongside the main step-by-step method.

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Frequently Asked Questions

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