Integral of \( \displaystyle \sin{\left(x \right)} \cos^{3}{\left(x \right)} \)
Problem 4.189 · medium Video
Find \( \displaystyle \int \sin{\left(x \right)} \cos^{3}{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx \]integral algebraStart with the integral of the given function. Rewrite the integrand to prepare for substitution.✓ Proved
- \[ = \int \frac{\sin{\left(2 x \right)} \cos^{2}{\left(x \right)}}{2}\, dx \]trig-identity linearityUse the double angle identity sin(2x) = 2sin(x)cos(x). Pull the constant factor out of the integral.✓ Proved
- \[ = \int \left(\frac{\cos{\left(2 x \right)}}{4} + \frac{1}{4}\right) \sin{\left(2 x \right)}\, dx \]trig-identityUse the power reduction identity for cos(x)**2.✓ Proved
- \[ = \int \left(\frac{\sin{\left(2 x \right)} \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{4}\right)\, dx \]algebraDistribute the terms inside the integral.✓ Proved
- \[ = \int \frac{\sin{\left(2 x \right)} \cos{\left(2 x \right)}}{4}\, dx + \int \frac{\sin{\left(2 x \right)}}{4}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \int \frac{\sin{\left(2 x \right)}}{4}\, dx + \int \frac{\sin{\left(4 x \right)}}{8}\, dx \]trig-identityUse the double angle identity again for the second term.✓ Proved
- \[ = - \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{32} \]antiderivativeEvaluate the remaining integrals.✓ Proved
Answer \( - \frac{\cos^{4}{\left(x \right)}}{4} + C \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -1/16 |
| 6 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -1/16 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | lines differ by the constant 1/32 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: lines differ by the constant 3/32 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 4 claims to use 'linearity' to pull out a constant, but the integral sign remains in the expression, meaning the constant was not actually pulled out. Furthermore, Step 5 applies a trigonometric identity to cos(x)^2, but the argument in the previous steps suggests a mix-up between x and 2x arguments, leading to an incorrect transformation in Step 5 where cos(x)^2 is replaced by (1+cos(2x))/2 inside an integral that already contains sin(2x), effectively changing the variable of integration context without proper substitution. Specifically, Step 5 replaces cos(x)^2 with (1+cos(2x))/2, but the term is multiplied by sin(2x). This is algebraically valid, but the label 'trig-identity' is correct. However, looking closely at Step 3 to 4: Step 3 is Integral(cos(x)**2 * (sin(2*x)/2), x). Step 4 is Integral(1/2 * cos(x)**2 * sin(2*x), x). This is just reordering constants, which is 'algebra' or 'simplify', not 'linearity' (which usually implies splitting or pulling out). But the real error is in Step 5. Step 5 transforms cos(x)^2 into (1+cos(2x))/2. This is a valid identity. But wait, look at Step 2 to 3. Step 2: cos(x)**2 * (cos(x)*sin(x)). Step 3: cos(x)**2 * (sin(2*x)/2). This is valid. Step 4: 1/2 * cos(x)**2 * sin(2*x). Valid. Step 5: 1/2 * (1/2 * (1 + cos(2*x))) * sin(2*x). This replaces cos(x)^2 with (1+cos(2x))/2. This is valid. Step 6: 1/4 * (sin(2*x) + cos(2*x)*sin(2*x)). Valid. Step 7: Split. Valid. Step 8: Second term 1/4 * cos(2x)sin(2x) becomes 1/8 * sin(4x). Valid. Step 9: Integrates 1/4 sin(2x) to -1/8 cos(2x) and 1/8 sin(4x) to -1/32 cos(4x). Valid. Wait, let's re-read the contract. 'Each step changes ONE thing.' Step 4 label is 'linearity'. Linearity usually refers to splitting sums or pulling constants out of the integral sign. Here, the integral sign is NOT removed. The constant 1/2 is just moved. This is 'algebra' or 'simplify'. Calling it 'linearity' is a mislabel if 'linearity' implies the integral operator's linearity property (pulling out). If 'linearity' is used for reordering constants, it's a stretch. Let's look at Step 5. It applies `cos(x)^2 = (1+cos(2x))/2`. This is a `trig-identity`. Correct. Let's look at Step 9. It integrates two terms at once. `Integral(A) + Integral(B) -> Antideriv(A) + Antideriv(B)`. This applies the antiderivative rule to two separate terms simultaneously. The contract says 'Each step changes ONE thing'. Applying the antiderivative to both terms in one step is applying the rule twice. This is a defect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 claims to use 'linearity' to pull out a constant, but the integral sign remains in the expression, meaning the constant was not actually pulled out. Furthermore, Step 5 applies a trigonometric identity to cos(x)^2, but the argument in the previous steps suggests a mix-up between x and 2x arguments, leading to an incorrect transformation in Step 5 where cos(x)^2 is replaced by (1+cos(2x))/2 inside an integral that already contains sin(2x), effectively changing the variable of integration context without proper substitution. Specifically, Step 5 replaces cos(x)^2 with (1+cos(2x))/2, but the term is multiplied by sin(2x). This is algebraically valid, but the label 'trig-identity' is correct. However, looking closely at Step 3 to 4: Step 3 is Integral(cos(x)**2 * (sin(2*x)/2), x). Step 4 is Integral(1/2 * cos(x)**2 * sin(2*x), x). This is just reordering constants, which is 'algebra' or 'simplify', not 'linearity' (which usually implies splitting or pulling out). But the real error is in Step 5. Step 5 transforms cos(x)^2 into (1+cos(2x))/2. This is a valid identity. But wait, look at Step 2 to 3. Step 2: cos(x)**2 * (cos(x)*sin(x)). Step 3: cos(x)**2 * (sin(2*x)/2). This is valid. Step 4: 1/2 * cos(x)**2 * sin(2*x). Valid. Step 5: 1/2 * (1/2 * (1 + cos(2*x))) * sin(2*x). This replaces cos(x)^2 with (1+cos(2x))/2. This is valid. Step 6: 1/4 * (sin(2*x) + cos(2*x)*sin(2*x)). Valid. Step 7: Split. Valid. Step 8: Second term 1/4 * cos(2x)sin(2x) becomes 1/8 * sin(4x). Valid. Step 9: Integrates 1/4 sin(2x) to -1/8 cos(2x) and 1/8 sin(4x) to -1/32 cos(4x). Valid. Wait, let's re-read the contract. 'Each step changes ONE thing.' Step 4 label is 'linearity'. Linearity usually refers to splitting sums or pulling constants out of the integral sign. Here, the integral sign is NOT removed. The constant 1/2 is just moved. This is 'algebra' or 'simplify'. Calling it 'linearity' is a mislabel if 'linearity' implies the integral operator's linearity property (pulling out). If 'linearity' is used for reordering constants, it's a stretch. Let's look at Step 5. It applies `cos(x)^2 = (1+cos(2x))/2`. This is a `trig-identity`. Correct. Let's look at Step 9. It integrates two terms at once. `Integral(A) + Integral(B) -> Antideriv(A) + Antideriv(B)`. This applies the antiderivative rule to two separate terms simultaneously. The contract says 'Each step changes ONE thing'. Applying the antiderivative to both terms in one step is applying the rule twice. This is a defect.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.