Home›Calculus 1›u-substitution›Problem 4.1129

Integral of \( \displaystyle \frac{\sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} \)

Problem 4.1129 · medium Video

Find \( \displaystyle \int \frac{\sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} \, dx \). (Omit the constant of integration.)

\[ \int \frac{\sin{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{\tan{\left(3 x \right)}}{\cos{\left(3 x \right)}}\, dx \]
rewriteRewrite the denominator as a power of cosine.✓ Proved
\[ = \int \tan{\left(3 x \right)} \sec{\left(3 x \right)}\, dx \]
rewrite simplifyUse the identity 1/cos(3*x) = sec(3*x). Simplify the integrand.✓ Proved
\[ = \frac{\sec{\left(3 x \right)}}{3} \]
antiderivativeApply the antiderivative rule for sec(ax)tan(ax).✓ Proved

Answer \( \frac{1}{3 \cos{\left(3 x \right)}} + C \)

Top: the graph of F = 1/(3cos(3x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(3x)/cos(3x)².

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 undefined where cos(3*x) = 0 tan has poles at odd multiples of pi/2
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(3*x) = 0 sec has poles at odd multiples of pi/2
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(3*x) = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated the stated antiderivative back to the integrand

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies trigonometric identities to transform the integrand into a standard form and then applies the antiderivative rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies substitution and standard antiderivative rules with appropriate labels. Each step changes only one aspect of the expression.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.