prompt:
Given f(x) = cos(x), compute -4★-3.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

problem_type: compute

f: cos(x)

correct_answer: 5.102077496632204


prompt:
Given f(x) = 1/x, compute 1★3.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.09861228866811


prompt:
Given f(x) = cos(x), compute -3◆5.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.182195733396728


prompt:
Given f(x) = -2x, compute 0★2.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 2.0


prompt:
Given f(x) = 1/x, compute -4◆-1.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.613705638880109


prompt:
Given f(x) = 1x, compute 3◆4.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 9.5


prompt:
Given f(x) = x^3, compute -3●-2.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -10.25


prompt:
Given f(x) = 3x, compute -5◆3.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -18.0


prompt:
Given f(x) = x^4, compute -3◆3.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 103.2


prompt:
Given f(x) = sin(x), compute -3●0.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

f: sin(x)

correct_answer: 4.010007503399555


prompt:
Given f(x) = sin(x), compute -3★-1.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.469705197531415


prompt:
Given f(x) = cos(x), compute 1◆4.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.401726519884175


prompt:
Given f(x) = cos(x), compute -5●-3.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.8999557172769945


prompt:
Given f(x) = sin(x), compute -5●3.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.273654682063672


prompt:
Given f(x) = cos(x), compute -5●-1.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.199604740528965


prompt:
Given f(x) = 1/x, a★1 = 20. Find a.
The operation a★b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].

problem_type: solve

known_var: b

target_var: a


prompt:
Given f(x) = sin(x), compute 2◆3.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.573845660053303


prompt:
Given f(x) = 1/x, compute -3◆-2.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.594534891891835


prompt:
Given f(x) = cos(x), compute -3★0.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.141120008059867


prompt:
Given f(x) = sin(x), compute -3◆0.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.010007503399555


prompt:
Given f(x) = x^3, compute -4★2.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -54.0


prompt:
Given f(x) = x^3, compute -5★1.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -150.0


prompt:
Given f(x) = 1/x, b●3 = 3. Find b.
The operation a●b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].

known_var: a

target_var: b


prompt:
Given f(x) = 1/x, compute 1●4.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.386294361119891


prompt:
Given f(x) = sin(x), compute -4★4.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.0


prompt:
Given f(x) = -2x, compute 0★3.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -3.0


prompt:
Given f(x) = cos(x), compute -1★0.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.841470984807897


prompt:
Given f(x) = 3x, compute 2●3.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 13.5


prompt:
Given f(x) = -1x, compute -5●-2.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 16.5


prompt:
Given f(x) = cos(x), compute -2◆-1.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.067826442017785


prompt:
Given f(x) = sin(x), compute 0★3.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.989992496600445


prompt:
Given f(x) = 1/x, compute -4●-3.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.712317927548219


prompt:
Given f(x) = 1/x, b★2 = 17. Find b.
The operation a★b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].


prompt:
Given f(x) = 2x, a★4 = 6. Find a.
The operation a★b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].


prompt:
Given f(x) = cos(x), compute 3●4.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.102077496632204


prompt:
Given f(x) = cos(x), compute 0●2.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.909297426825682


prompt:
Given f(x) = 1/x, compute -5★-4.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.77685644868579


prompt:
Given f(x) = sin(x), compute -5★1.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.743359879595086


prompt:
Given f(x) = 1/x, compute 2◆5.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.916290731874155


prompt:
Given f(x) = -2x, compute 4◆5.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -3.0


prompt:
Given f(x) = 1/x, compute 1◆5.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.6094379124341005


prompt:
Given f(x) = 1/x, compute -4●-1.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.613705638880109


prompt:
Given f(x) = sin(x), compute 2●5.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.300190977989631


prompt:
Given f(x) = sin(x), compute 1●4.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.193945926731752


prompt:
Given f(x) = cos(x), compute -4●5.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.2842732300289335


prompt:
Given f(x) = 1/x, compute 3●5.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.5108256237659905


prompt:
Given f(x) = sin(x), compute 1◆4.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.193945926731752


prompt:
Given f(x) = x^2, a●0 = 17. Find a.
The operation a●b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].


prompt:
Given f(x) = x^3, compute -3●3.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.0


prompt:
Given f(x) = x^3, compute -4◆2.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: -54.0


prompt:
Given f(x) = 1x, compute -1★5.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 18.0


prompt:
Given f(x) = cos(x), compute -2●0.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.909297426825682


prompt:
Given f(x) = 1/x, compute -2◆-1.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 5.306852819440055


prompt:
Given f(x) = cos(x), compute -5◆-3.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.8999557172769945


prompt:
Given f(x) = cos(x), compute -3★0.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.141120008059867


prompt:
Given f(x) = -2x, compute -1◆0.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 7.0


prompt:
Given f(x) = cos(x), compute -4★5.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 4.2842732300289335


prompt:
Given f(x) = 1/x, compute 3◆5.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.5108256237659905


prompt:
Given f(x) = x^2, compute 3◆4.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 18.333333333333332


prompt:
Given f(x) = sin(x), compute 0★5.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.716337814536773


prompt:
Given f(x) = 1/x, compute 3★5.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.5108256237659905


prompt:
Given f(x) = x^3, a◆-3 = 3. Find a.
The operation a◆b is defined as ∫_a^b f(x)dx + 6. For multiple answers, separate with 'or'. Use fractions if needed. Put your answer in [[ ]].


prompt:
Given f(x) = -2x, compute -5●2.
The operation a●b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 27.0


prompt:
Given f(x) = 1/x, compute 3★5.
The operation a★b is defined as ∫_a^b f(x)dx + 6. Calculate the result. For fractions, use 'a/b' format. Put your answer in [[ ]].

correct_answer: 6.5108256237659905