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Trigonometric integrals

Integrals of sin(x) and cos(x): Signs and Factors

These antiderivatives reverse the derivative cycle. The negative sign belongs to the sine integral because cosine differentiates to negative sine.

Direct answer

∫sin(x)dx=−cos(x)+C; ∫cos(x)dx=sin(x)+C

For sin(ax+b) or cos(ax+b), divide by the nonzero inner slope a.

When to use integrals of sine and cosine

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Use direct trigonometric antiderivatives when the inner derivative is present up to a constant.

Ask first 2

Which condition can change the answer?

What to do

For sin(ax+b) or cos(ax+b), divide by the nonzero inner slope a.

Ask first 3

How should the result be checked?

What to do

Differentiate the answer and compare both function and coefficient.

Reverse the derivative cycle

Differentiate −cos(x) to recover sin(x).

Differentiate sin(x) to recover cos(x).

For ax+b, substitution introduces division by a.

A nonlinear angle needs a matching derivative factor or another method.

1
[−cos(x)]'=sin(x)
2
[sin(x)]'=cos(x)
3
∫sin(ax+b)dx=−cos(ax+b)/a+C

Worked examples

∫sin⁡x dx\int \sin x\,\mathrm{d}x
−cos⁡x+C-\cos x + C

Keep the negative sign.

∫cos⁡(3 x) dx\int \cos\left(3\,x\right)\,\mathrm{d}x
sin⁡(3 x)3+C\frac{\sin\left(3\,x\right)}{3} + C

Divide by the inner slope.

∫2x cos(x²) dx
sin⁡(x2)+C\sin\left(x^{2}\right) + C

The inner derivative is present.

Common mistakes

  • • Missing the sine negative sign.
  • • Multiplying by a instead of dividing.
  • • Assuming every composition is elementary.
  • • Omitting C.

Error clinic: locate the first invalid step

Incorrect attempt

∫sin(x)dx=cos(x)+C

Why it fails

cos′(x)=−sin(x).

Correction

−cos⁡x+C-\cos x + C

Incorrect attempt

∫cos(3x)dx=3sin(3x)+C

Why it fails

Its derivative is 9cos(3x).

Correction

sin⁡(3 x)3+C\frac{\sin\left(3\,x\right)}{3} + C

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

∫cos⁡(3 x) dx\int \cos\left(3\,x\right)\,\mathrm{d}x

Check before solving

Divide by the inner slope.

Reveal the reference answer

Reference answer

sin⁡(3 x)3+C\frac{\sin\left(3\,x\right)}{3} + C
∫2x cos(x²) dx

Check before solving

The inner derivative is present.

Reveal the reference answer

Reference answer

sin⁡(x2)+C\sin\left(x^{2}\right) + C

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

積分:sin⁡x\sin x積分:cos⁡(3 x)\cos\left(3\,x\right)
Open the calculator without a preset: 積分

Frequently asked questions

Why negative cosine?+

Differentiating −cos(x) gives sin(x).

Why divide by a?+

Differentiation multiplies by a, so integration compensates.

What if the inner derivative is missing?+

A constant can be adjusted; a nonconstant missing factor may require another method.

Conclusion

Reverse the derivative cycle, preserve the sine sign, adjust for the inner slope, and verify the answer.

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