Final Answer
$\frac{a^{\left(3x+1\right)}}{3x+1}-3a^{\left(2x+1\right)}b^x+\frac{3a^{\left(2x+1\right)}xb^x}{2x+1}+\frac{3a^{\left(2x+1\right)}}{2x+1}+\frac{1}{2}b^{3x}a^2\ln\left(b\right)-ab^{3x}+\frac{3a^{\left(x+1\right)}b^{2x}}{x+1}-\frac{3}{2}b^{2x}a^2\ln\left(b\right)+3ab^{2x}+a+\frac{3a^{\left(x+1\right)}}{x+1}+\frac{-3b^x}{\ln\left(b\right)}+\frac{-6a^{\left(x+1\right)}b^x}{x+1}+C_0$
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Step-by-step Solution
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Find the integral
$\int\left(a^x-b^x+1\right)^3da$
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$\int\left(a^x-b^x+1\right)^3da$
Learn how to solve problems step by step online. Find the integral of (a^x-b^x+1)^3. Find the integral. Rewrite the integrand \left(a^x-b^x+1\right)^3 in expanded form. Expand the integral \int\left(a^{3x}-3a^{2x}b^x+3a^{2x}-b^{3x}+3a^xb^{2x}+3b^{2x}+1+3a^x-3b^x-6a^xb^x\right)da into 10 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int-3a^{2x}b^xda results in: -3\int a^{2x}b^xda.
Final Answer
$\frac{a^{\left(3x+1\right)}}{3x+1}-3a^{\left(2x+1\right)}b^x+\frac{3a^{\left(2x+1\right)}xb^x}{2x+1}+\frac{3a^{\left(2x+1\right)}}{2x+1}+\frac{1}{2}b^{3x}a^2\ln\left(b\right)-ab^{3x}+\frac{3a^{\left(x+1\right)}b^{2x}}{x+1}-\frac{3}{2}b^{2x}a^2\ln\left(b\right)+3ab^{2x}+a+\frac{3a^{\left(x+1\right)}}{x+1}+\frac{-3b^x}{\ln\left(b\right)}+\frac{-6a^{\left(x+1\right)}b^x}{x+1}+C_0$