Integrate the following to find an expression for y:
Evaluate the following integrals:
Evaluate the following integrals:
\int 5x\,(2x^{2}+3)^{2}\, dx
\int 3x^2\,(x^{3}+1)^{5}\, dx
\int 2x^{3}\,(2+8x^{4})^{2}\, dx
\int 2x\,(5+3x^{2})^{3}\, dx
\int x\,(2-x^{2})^{2}\, dx
\int 2x^{2}\,\sqrt{x^{3}-1}\, dx
\int 5x\,\sqrt{3x^{2}+2}\, dx
\int \dfrac{4x}{\sqrt{x^{2}-4}}\, dx
\int \dfrac{x-6}{\sqrt{x^{2}-12x+5}}\, dx
\int \dfrac{4x}{({x^{2}+3})^{3}}\, dx
Find:
\int 10x^{2} (x^{3}-1)^{4}\, dx
\int \dfrac{7x}{\sqrt{x^{2}+4}}\, dx
\int \dfrac{5(\sqrt{x}-1)^{2}}{3\sqrt{x}}\, dx
\int 28(9x^{2}+12x)(x^{3}+2x^{2}-3)^{3}\, dx
\int (12x-12)(7-x^{2}+2x)^{5}\, dx
Consider the function y=x^{5} + x^{4}.
Find \dfrac{d}{dx} \left(x^{5} + x^{4}\right).
Hence, determine \int \left( 35 x^{4} + 28 x^{3}\right)\, dx.
Consider the function y=x^{ - 4 }.
Find \dfrac{d}{dx} \left(x^{ - 4 }\right).
Hence, determine \int 16 x^{ - 5 }\, dx.
Consider the function f(x) = x^{6}.
Find \dfrac{d}{dx} \left(x^{6}\right).
Hence, determine \int 24 x^{5}\, dx.
Consider the function f(x) = x^{4}.
Find \dfrac{d}{dx} \left(x^{4}\right).
Hence, determine \int 4 x^{3}\, dx.
Consider the function y = x^{8} + x^{6}.
Find \dfrac{d}{dx} \left(x^{8} + x^{6}\right).
Hence, determine \int \left( 4 x^{7} + 3 x^{5}\right)\, dx.
Consider the function f(x) = \sqrt[5]{x^{6}}.
Find \dfrac{d}{dx} \left(\sqrt[5]{x^{6}}\right).
Hence, determine \int 6 \sqrt[5]{x}\, dx.
Consider the function y = \left(x^{2} - 3\right)^{5}.
Find \dfrac{d}{dx} \left(x^{2} - 3\right)^{5}.
Hence, determine \int 30 x \left(x^{2} - 3\right)^{4}\, dx.
Consider the function f(x) = \left( 5 x^{2} + 10 x - 3\right)^{5}.
Find \dfrac{d}{dx} \left( 5 x^{2} + 10 x - 3\right)^{5}.
Hence, find \int 150 \left(x + 1\right) \left( 5 x^{2} + 10 x - 3\right)^{4}\, dx.
Consider the function y = \left(x^{2} - 3 x + 6\right)^{5}.
Find \dfrac{d}{dx} \left(x^{2} - 3 x + 6\right)^{5}.
Hence, find \int \left(3 - 2 x\right) \left(x^{2} - 3 x + 6\right)^{4}\, dx.
Consider the function f(x) = \sqrt{ 4 x + 11}.
Find \dfrac{d}{dx} \left(\sqrt{ 4 x + 11}\right).
Hence, determine \int \dfrac{12}{\sqrt{ 4 x + 11}}\, dx.
Consider the function y = \dfrac{1}{\left(x^{2} + 7\right)^{3}}.
Find \dfrac{d}{dx} \left(\dfrac{1}{\left(x^{2} + 7\right)^{3}}\right).
Hence, determine \int - \dfrac{24 x}{\left(x^{2} + 7\right)^{4}} \, dx.