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In this paper, we establish new <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><msub><mrow></mrow><msub><mi>κ</mi><mn>1</mn></msub></msub></mrow></semantics></math></inline-formula>-integral and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><msup><mrow></mrow><msub><mi>κ</mi><mn>2</mn></msub></msup></mrow></semantics></math></inline-formula>-integral identities. By employing these new identities, we establish new <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><msub><mrow></mrow><msub><mi>κ</mi><mn>1</mn></msub></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><msup><mrow></mrow><msub><mi>κ</mi><mn>2</mn></msub></msup></mrow></semantics></math></inline-formula>- trapezoidal integral-type inequalities through strongly convex and quasi-convex functions. Finally, some examples are given to illustrate the investigated results.